[{"data":1,"prerenderedAt":17418},["ShallowReactive",2],{"note:math\u002Fabstract_algebra\u002Fsubringideal":3,"site-search-catalogue":17417},{"id":4,"title":5,"body":6,"description":17381,"extension":17409,"meta":17410,"navigation":17412,"path":17413,"seo":17414,"stem":17415,"__hash__":17416},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fsubringideal.md","子环、理想与商环",{"type":7,"value":8,"toc":17380},"minimark",[9,13,18,22,98,101,105,110,446,727,729,733,737,740,802,1347,1449,1453,1456,1599,1769,5258,5263,6284,6286,6290,6294,6628,6908,6912,6915,6920,7965,8466,8534,9010,9047,9215,9659,9830,9832,9836,9911,9967,10026,11201,11257,11500,13313,13351,13825,14789,14791,14795,14799,14838,15328,16567,16571,16643],[10,11,5],"h1",{"id":12},"子环理想与商环",[14,15,17],"h2",{"id":16},"_1-课程总览群论与环论的类比","1. 课程总览：群论与环论的类比",[19,20,21],"p",{},"环论的学习可以类比群论。通过对比，我们可以更直观地理解新概念。",[23,24,25,42],"table",{},[26,27,28],"thead",{},[29,30,31,36,39],"tr",{},[32,33,35],"th",{"align":34},"left","群论概念 (Group Theory)",[32,37,38],{"align":34},"环论对应概念 (Ring Theory)",[32,40,41],{"align":34},"备注",[43,44,45,64,81],"tbody",{},[29,46,47,55,61],{},[48,49,50,54],"td",{"align":34},[51,52,53],"strong",{},"子群"," (Subgroup)",[48,56,57,60],{"align":34},[51,58,59],{},"子环"," (Subring)",[48,62,63],{"align":34},"性质相似",[29,65,66,72,78],{},[48,67,68,71],{"align":34},[51,69,70],{},"正规子群"," (Normal Subgroup)",[48,73,74,77],{"align":34},[51,75,76],{},"理想"," (Ideal)",[48,79,80],{"align":34},"都在构建商结构时起关键作用",[29,82,83,89,95],{},[48,84,85,88],{"align":34},[51,86,87],{},"商群"," (Quotient Group)",[48,90,91,94],{"align":34},[51,92,93],{},"商环"," (Quotient Ring)",[48,96,97],{"align":34},"由理想对应的陪集构成的环",[99,100],"hr",{},[14,102,104],{"id":103},"_2-子环-subring","2. 子环 (Subring)",[106,107,109],"h3",{"id":108},"_21-定义与基本性质","2.1 定义与基本性质",[19,111,112,113,209,210,268,269,297,298,326,327,330,331,334,335,363,364,392,393,445],{},"设 ",[114,115,118,161],"span",{"className":116},[117],"katex",[114,119,122],{"className":120},[121],"katex-mathml",[123,124,126],"math",{"xmlns":125},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[127,128,129,156],"semantics",{},[130,131,132,137,141,145,148,150,153],"mrow",{},[133,134,136],"mo",{"stretchy":135},"false","(",[138,139,140],"mi",{},"R",[133,142,144],{"separator":143},"true",",",[133,146,147],{},"+",[133,149,144],{"separator":143},[133,151,152],{},"⋅",[133,154,155],{"stretchy":135},")",[157,158,160],"annotation",{"encoding":159},"application\u002Fx-tex","(R, +, \\cdot)",[114,162,165],{"className":163,"ariaHidden":143},[164],"katex-html",[114,166,169,174,178,184,188,193,196,199,202,205],{"className":167},[168],"base",[114,170],{"className":171,"style":173},[172],"strut","height:1em;vertical-align:-0.25em;",[114,175,136],{"className":176},[177],"mopen",[114,179,140],{"className":180,"style":183},[181,182],"mord","mathnormal","margin-right:0.0077em;",[114,185,144],{"className":186},[187],"mpunct",[114,189],{"className":190,"style":192},[191],"mspace","margin-right:0.1667em;",[114,194,147],{"className":195},[181],[114,197,144],{"className":198},[187],[114,200],{"className":201,"style":192},[191],[114,203,152],{"className":204},[181],[114,206,155],{"className":207},[208],"mclose"," 是一个环，",[114,211,213,233],{"className":212},[117],[114,214,216],{"className":215},[121],[123,217,218],{"xmlns":125},[127,219,220,230],{},[130,221,222,225,228],{},[138,223,224],{},"S",[133,226,227],{},"⊆",[138,229,140],{},[157,231,232],{"encoding":159},"S \\subseteq R",[114,234,236,258],{"className":235,"ariaHidden":143},[164],[114,237,239,243,247,251,255],{"className":238},[168],[114,240],{"className":241,"style":242},[172],"height:0.8193em;vertical-align:-0.136em;",[114,244,224],{"className":245,"style":246},[181,182],"margin-right:0.0576em;",[114,248],{"className":249,"style":250},[191],"margin-right:0.2778em;",[114,252,227],{"className":253},[254],"mrel",[114,256],{"className":257,"style":250},[191],[114,259,261,265],{"className":260},[168],[114,262],{"className":263,"style":264},[172],"height:0.6833em;",[114,266,140],{"className":267,"style":183},[181,182]," 是其子集。\n如果 ",[114,270,272,285],{"className":271},[117],[114,273,275],{"className":274},[121],[123,276,277],{"xmlns":125},[127,278,279,283],{},[130,280,281],{},[138,282,224],{},[157,284,224],{"encoding":159},[114,286,288],{"className":287,"ariaHidden":143},[164],[114,289,291,294],{"className":290},[168],[114,292],{"className":293,"style":264},[172],[114,295,224],{"className":296,"style":246},[181,182]," 在 ",[114,299,301,314],{"className":300},[117],[114,302,304],{"className":303},[121],[123,305,306],{"xmlns":125},[127,307,308,312],{},[130,309,310],{},[138,311,140],{},[157,313,140],{"encoding":159},[114,315,317],{"className":316,"ariaHidden":143},[164],[114,318,320,323],{"className":319},[168],[114,321],{"className":322,"style":264},[172],[114,324,140],{"className":325,"style":183},[181,182]," 原有的",[51,328,329],{},"加法","和",[51,332,333],{},"乘法","运算下依然构成一个环，则称 ",[114,336,338,351],{"className":337},[117],[114,339,341],{"className":340},[121],[123,342,343],{"xmlns":125},[127,344,345,349],{},[130,346,347],{},[138,348,224],{},[157,350,224],{"encoding":159},[114,352,354],{"className":353,"ariaHidden":143},[164],[114,355,357,360],{"className":356},[168],[114,358],{"className":359,"style":264},[172],[114,361,224],{"className":362,"style":246},[181,182]," 是 ",[114,365,367,380],{"className":366},[117],[114,368,370],{"className":369},[121],[123,371,372],{"xmlns":125},[127,373,374,378],{},[130,375,376],{},[138,377,140],{},[157,379,140],{"encoding":159},[114,381,383],{"className":382,"ariaHidden":143},[164],[114,384,386,389],{"className":385},[168],[114,387],{"className":388,"style":264},[172],[114,390,140],{"className":391,"style":183},[181,182]," 的子环，记为 ",[114,394,396,415],{"className":395},[117],[114,397,399],{"className":398},[121],[123,400,401],{"xmlns":125},[127,402,403,412],{},[130,404,405,407,410],{},[138,406,224],{},[133,408,409],{},"≤",[138,411,140],{},[157,413,414],{"encoding":159},"S \\le R",[114,416,418,436],{"className":417,"ariaHidden":143},[164],[114,419,421,424,427,430,433],{"className":420},[168],[114,422],{"className":423,"style":242},[172],[114,425,224],{"className":426,"style":246},[181,182],[114,428],{"className":429,"style":250},[191],[114,431,409],{"className":432},[254],[114,434],{"className":435,"style":250},[191],[114,437,439,442],{"className":438},[168],[114,440],{"className":441,"style":264},[172],[114,443,140],{"className":444,"style":183},[181,182],"。",[447,448,449,514],"ul",{},[450,451,452,455,456,484,485,513],"li",{},[51,453,454],{},"性质","：如果 ",[114,457,459,472],{"className":458},[117],[114,460,462],{"className":461},[121],[123,463,464],{"xmlns":125},[127,465,466,470],{},[130,467,468],{},[138,469,140],{},[157,471,140],{"encoding":159},[114,473,475],{"className":474,"ariaHidden":143},[164],[114,476,478,481],{"className":477},[168],[114,479],{"className":480,"style":264},[172],[114,482,140],{"className":483,"style":183},[181,182]," 是交换环，则子环 ",[114,486,488,501],{"className":487},[117],[114,489,491],{"className":490},[121],[123,492,493],{"xmlns":125},[127,494,495,499],{},[130,496,497],{},[138,498,224],{},[157,500,224],{"encoding":159},[114,502,504],{"className":503,"ariaHidden":143},[164],[114,505,507,510],{"className":506},[168],[114,508],{"className":509,"style":264},[172],[114,511,224],{"className":512,"style":246},[181,182]," 也是交换环。",[450,515,516,519,520],{},[51,517,518],{},"注意（与群论的区别）","：\n",[447,521,522,525,530],{},[450,523,524],{},"群论中，子群的单位元一定等于大群的单位元。",[450,526,527],{},[51,528,529],{},"环论中，子环的单位元可能与大环不同，甚至可能没有单位元。",[450,531,532,536,537,363,577,606,607,363,675,726],{},[533,534,535],"em",{},"例","：",[114,538,540,560],{"className":539},[117],[114,541,543],{"className":542},[121],[123,544,545],{"xmlns":125},[127,546,547,557],{},[130,548,549,553],{},[550,551,552],"mn",{},"2",[138,554,556],{"mathvariant":555},"double-struck","Z",[157,558,559],{"encoding":159},"2\\mathbb{Z}",[114,561,563],{"className":562,"ariaHidden":143},[164],[114,564,566,570,573],{"className":565},[168],[114,567],{"className":568,"style":569},[172],"height:0.6889em;",[114,571,552],{"className":572},[181],[114,574,556],{"className":575},[181,576],"mathbb",[114,578,580,594],{"className":579},[117],[114,581,583],{"className":582},[121],[123,584,585],{"xmlns":125},[127,586,587,591],{},[130,588,589],{},[138,590,556],{"mathvariant":555},[157,592,593],{"encoding":159},"\\mathbb{Z}",[114,595,597],{"className":596,"ariaHidden":143},[164],[114,598,600,603],{"className":599},[168],[114,601],{"className":602,"style":569},[172],[114,604,556],{"className":605},[181,576]," 的子环，但无单位元；",[114,608,610,636],{"className":609},[117],[114,611,613],{"className":612},[121],[123,614,615],{"xmlns":125},[127,616,617,633],{},[130,618,619,621,624,627,630],{},[138,620,556],{"mathvariant":555},[133,622,623],{},"⊕",[133,625,626],{"stretchy":135},"{",[550,628,629],{},"0",[133,631,632],{"stretchy":135},"}",[157,634,635],{"encoding":159},"\\mathbb{Z} \\oplus \\{0\\}",[114,637,639,660],{"className":638,"ariaHidden":143},[164],[114,640,642,646,649,653,657],{"className":641},[168],[114,643],{"className":644,"style":645},[172],"height:0.7722em;vertical-align:-0.0833em;",[114,647,556],{"className":648},[181,576],[114,650],{"className":651,"style":652},[191],"margin-right:0.2222em;",[114,654,623],{"className":655},[656],"mbin",[114,658],{"className":659,"style":652},[191],[114,661,663,666,669,672],{"className":662},[168],[114,664],{"className":665,"style":173},[172],[114,667,626],{"className":668},[177],[114,670,629],{"className":671},[181],[114,673,632],{"className":674},[208],[114,676,678,696],{"className":677},[117],[114,679,681],{"className":680},[121],[123,682,683],{"xmlns":125},[127,684,685,693],{},[130,686,687,689,691],{},[138,688,556],{"mathvariant":555},[133,690,623],{},[138,692,556],{"mathvariant":555},[157,694,695],{"encoding":159},"\\mathbb{Z} \\oplus \\mathbb{Z}",[114,697,699,717],{"className":698,"ariaHidden":143},[164],[114,700,702,705,708,711,714],{"className":701},[168],[114,703],{"className":704,"style":645},[172],[114,706,556],{"className":707},[181,576],[114,709],{"className":710,"style":652},[191],[114,712,623],{"className":713},[656],[114,715],{"className":716,"style":652},[191],[114,718,720,723],{"className":719},[168],[114,721],{"className":722,"style":569},[172],[114,724,556],{"className":725},[181,576]," 的子环，单位元不同。",[99,728],{},[14,730,732],{"id":731},"_3-理想-ideal","3. 理想 (Ideal)",[106,734,736],{"id":735},"_31-定义左理想右理想与双侧理想","3.1 定义：左理想、右理想与双侧理想",[19,738,739],{},"在非交换环中，乘法有左右之分，因此必须严格区分乘法的“吸收”方向。",[19,741,112,742,772,773,801],{},[114,743,745,759],{"className":744},[117],[114,746,748],{"className":747},[121],[123,749,750],{"xmlns":125},[127,751,752,757],{},[130,753,754],{},[138,755,756],{},"I",[157,758,756],{"encoding":159},[114,760,762],{"className":761,"ariaHidden":143},[164],[114,763,765,768],{"className":764},[168],[114,766],{"className":767,"style":264},[172],[114,769,756],{"className":770,"style":771},[181,182],"margin-right:0.0785em;"," 是环 ",[114,774,776,789],{"className":775},[117],[114,777,779],{"className":778},[121],[123,780,781],{"xmlns":125},[127,782,783,787],{},[130,784,785],{},[138,786,140],{},[157,788,140],{"encoding":159},[114,790,792],{"className":791,"ariaHidden":143},[164],[114,793,795,798],{"className":794},[168],[114,796],{"className":797,"style":264},[172],[114,799,140],{"className":800,"style":183},[181,182]," 的一个子环（首先它是加法子群）：",[803,804,805,1023,1228],"ol",{},[450,806,807,810,811,903,904,960,961,989,990,993,994,1022],{},[51,808,809],{},"左理想 (Left Ideal)","：\n对任意 ",[114,812,814,843],{"className":813},[117],[114,815,817],{"className":816},[121],[123,818,819],{"xmlns":125},[127,820,821,840],{},[130,822,823,826,829,831,833,836,838],{},[138,824,825],{},"r",[133,827,828],{},"∈",[138,830,140],{},[133,832,144],{"separator":143},[138,834,835],{},"x",[133,837,828],{},[138,839,756],{},[157,841,842],{"encoding":159},"r \\in R, x \\in I",[114,844,846,866,894],{"className":845,"ariaHidden":143},[164],[114,847,849,853,857,860,863],{"className":848},[168],[114,850],{"className":851,"style":852},[172],"height:0.5782em;vertical-align:-0.0391em;",[114,854,825],{"className":855,"style":856},[181,182],"margin-right:0.0278em;",[114,858],{"className":859,"style":250},[191],[114,861,828],{"className":862},[254],[114,864],{"className":865,"style":250},[191],[114,867,869,873,876,879,882,885,888,891],{"className":868},[168],[114,870],{"className":871,"style":872},[172],"height:0.8778em;vertical-align:-0.1944em;",[114,874,140],{"className":875,"style":183},[181,182],[114,877,144],{"className":878},[187],[114,880],{"className":881,"style":192},[191],[114,883,835],{"className":884},[181,182],[114,886],{"className":887,"style":250},[191],[114,889,828],{"className":890},[254],[114,892],{"className":893,"style":250},[191],[114,895,897,900],{"className":896},[168],[114,898],{"className":899,"style":264},[172],[114,901,756],{"className":902,"style":771},[181,182],"，都有 ",[114,905,907,927],{"className":906},[117],[114,908,910],{"className":909},[121],[123,911,912],{"xmlns":125},[127,913,914,924],{},[130,915,916,918,920,922],{},[138,917,825],{},[138,919,835],{},[133,921,828],{},[138,923,756],{},[157,925,926],{"encoding":159},"rx \\in I",[114,928,930,951],{"className":929,"ariaHidden":143},[164],[114,931,933,936,939,942,945,948],{"className":932},[168],[114,934],{"className":935,"style":852},[172],[114,937,825],{"className":938,"style":856},[181,182],[114,940,835],{"className":941},[181,182],[114,943],{"className":944,"style":250},[191],[114,946,828],{"className":947},[254],[114,949],{"className":950,"style":250},[191],[114,952,954,957],{"className":953},[168],[114,955],{"className":956,"style":264},[172],[114,958,756],{"className":959,"style":771},[181,182],"。（",[114,962,964,977],{"className":963},[117],[114,965,967],{"className":966},[121],[123,968,969],{"xmlns":125},[127,970,971,975],{},[130,972,973],{},[138,974,140],{},[157,976,140],{"encoding":159},[114,978,980],{"className":979,"ariaHidden":143},[164],[114,981,983,986],{"className":982},[168],[114,984],{"className":985,"style":264},[172],[114,987,140],{"className":988,"style":183},[181,182]," 中的元素从",[51,991,992],{},"左边","乘进去，结果仍在 ",[114,995,997,1010],{"className":996},[117],[114,998,1000],{"className":999},[121],[123,1001,1002],{"xmlns":125},[127,1003,1004,1008],{},[130,1005,1006],{},[138,1007,756],{},[157,1009,756],{"encoding":159},[114,1011,1013],{"className":1012,"ariaHidden":143},[164],[114,1014,1016,1019],{"className":1015},[168],[114,1017],{"className":1018,"style":264},[172],[114,1020,756],{"className":1021,"style":771},[181,182]," 中）。",[450,1024,1025,810,1028,903,1113,960,1169,989,1197,993,1200,1022],{},[51,1026,1027],{},"右理想 (Right Ideal)",[114,1029,1031,1056],{"className":1030},[117],[114,1032,1034],{"className":1033},[121],[123,1035,1036],{"xmlns":125},[127,1037,1038,1054],{},[130,1039,1040,1042,1044,1046,1048,1050,1052],{},[138,1041,825],{},[133,1043,828],{},[138,1045,140],{},[133,1047,144],{"separator":143},[138,1049,835],{},[133,1051,828],{},[138,1053,756],{},[157,1055,842],{"encoding":159},[114,1057,1059,1077,1104],{"className":1058,"ariaHidden":143},[164],[114,1060,1062,1065,1068,1071,1074],{"className":1061},[168],[114,1063],{"className":1064,"style":852},[172],[114,1066,825],{"className":1067,"style":856},[181,182],[114,1069],{"className":1070,"style":250},[191],[114,1072,828],{"className":1073},[254],[114,1075],{"className":1076,"style":250},[191],[114,1078,1080,1083,1086,1089,1092,1095,1098,1101],{"className":1079},[168],[114,1081],{"className":1082,"style":872},[172],[114,1084,140],{"className":1085,"style":183},[181,182],[114,1087,144],{"className":1088},[187],[114,1090],{"className":1091,"style":192},[191],[114,1093,835],{"className":1094},[181,182],[114,1096],{"className":1097,"style":250},[191],[114,1099,828],{"className":1100},[254],[114,1102],{"className":1103,"style":250},[191],[114,1105,1107,1110],{"className":1106},[168],[114,1108],{"className":1109,"style":264},[172],[114,1111,756],{"className":1112,"style":771},[181,182],[114,1114,1116,1136],{"className":1115},[117],[114,1117,1119],{"className":1118},[121],[123,1120,1121],{"xmlns":125},[127,1122,1123,1133],{},[130,1124,1125,1127,1129,1131],{},[138,1126,835],{},[138,1128,825],{},[133,1130,828],{},[138,1132,756],{},[157,1134,1135],{"encoding":159},"xr \\in I",[114,1137,1139,1160],{"className":1138,"ariaHidden":143},[164],[114,1140,1142,1145,1148,1151,1154,1157],{"className":1141},[168],[114,1143],{"className":1144,"style":852},[172],[114,1146,835],{"className":1147},[181,182],[114,1149,825],{"className":1150,"style":856},[181,182],[114,1152],{"className":1153,"style":250},[191],[114,1155,828],{"className":1156},[254],[114,1158],{"className":1159,"style":250},[191],[114,1161,1163,1166],{"className":1162},[168],[114,1164],{"className":1165,"style":264},[172],[114,1167,756],{"className":1168,"style":771},[181,182],[114,1170,1172,1185],{"className":1171},[117],[114,1173,1175],{"className":1174},[121],[123,1176,1177],{"xmlns":125},[127,1178,1179,1183],{},[130,1180,1181],{},[138,1182,140],{},[157,1184,140],{"encoding":159},[114,1186,1188],{"className":1187,"ariaHidden":143},[164],[114,1189,1191,1194],{"className":1190},[168],[114,1192],{"className":1193,"style":264},[172],[114,1195,140],{"className":1196,"style":183},[181,182],[51,1198,1199],{},"右边",[114,1201,1203,1216],{"className":1202},[117],[114,1204,1206],{"className":1205},[121],[123,1207,1208],{"xmlns":125},[127,1209,1210,1214],{},[130,1211,1212],{},[138,1213,756],{},[157,1215,756],{"encoding":159},[114,1217,1219],{"className":1218,"ariaHidden":143},[164],[114,1220,1222,1225],{"className":1221},[168],[114,1223],{"className":1224,"style":264},[172],[114,1226,756],{"className":1227,"style":771},[181,182],[450,1229,1230,1233,1234,1262,1263,1291,1292,1294,1295,445],{},[51,1231,1232],{},"理想 (Ideal \u002F Two-sided Ideal)","：\n如果 ",[114,1235,1237,1250],{"className":1236},[117],[114,1238,1240],{"className":1239},[121],[123,1241,1242],{"xmlns":125},[127,1243,1244,1248],{},[130,1245,1246],{},[138,1247,756],{},[157,1249,756],{"encoding":159},[114,1251,1253],{"className":1252,"ariaHidden":143},[164],[114,1254,1256,1259],{"className":1255},[168],[114,1257],{"className":1258,"style":264},[172],[114,1260,756],{"className":1261,"style":771},[181,182]," 既是左理想又是右理想，称 ",[114,1264,1266,1279],{"className":1265},[117],[114,1267,1269],{"className":1268},[121],[123,1270,1271],{"xmlns":125},[127,1272,1273,1277],{},[130,1274,1275],{},[138,1276,756],{},[157,1278,756],{"encoding":159},[114,1280,1282],{"className":1281,"ariaHidden":143},[164],[114,1283,1285,1288],{"className":1284},[168],[114,1286],{"className":1287,"style":264},[172],[114,1289,756],{"className":1290,"style":771},[181,182]," 为",[51,1293,76],{},"，记为 ",[114,1296,1298,1317],{"className":1297},[117],[114,1299,1301],{"className":1300},[121],[123,1302,1303],{"xmlns":125},[127,1304,1305,1314],{},[130,1306,1307,1309,1312],{},[138,1308,756],{},[133,1310,1311],{},"◃",[138,1313,140],{},[157,1315,1316],{"encoding":159},"I \\triangleleft R",[114,1318,1320,1338],{"className":1319,"ariaHidden":143},[164],[114,1321,1323,1326,1329,1332,1335],{"className":1322},[168],[114,1324],{"className":1325,"style":264},[172],[114,1327,756],{"className":1328,"style":771},[181,182],[114,1330],{"className":1331,"style":652},[191],[114,1333,1311],{"className":1334},[656],[114,1336],{"className":1337,"style":652},[191],[114,1339,1341,1344],{"className":1340},[168],[114,1342],{"className":1343,"style":264},[172],[114,1345,140],{"className":1346,"style":183},[181,182],[1348,1349,1350],"blockquote",{},[19,1351,1352,1355,1356,1384,1385,1448],{},[51,1353,1354],{},"注","：若 ",[114,1357,1359,1372],{"className":1358},[117],[114,1360,1362],{"className":1361},[121],[123,1363,1364],{"xmlns":125},[127,1365,1366,1370],{},[130,1367,1368],{},[138,1369,140],{},[157,1371,140],{"encoding":159},[114,1373,1375],{"className":1374,"ariaHidden":143},[164],[114,1376,1378,1381],{"className":1377},[168],[114,1379],{"className":1380,"style":264},[172],[114,1382,140],{"className":1383,"style":183},[181,182]," 是交换环，则 ",[114,1386,1388,1411],{"className":1387},[117],[114,1389,1391],{"className":1390},[121],[123,1392,1393],{"xmlns":125},[127,1394,1395,1408],{},[130,1396,1397,1399,1401,1404,1406],{},[138,1398,825],{},[138,1400,835],{},[133,1402,1403],{},"=",[138,1405,835],{},[138,1407,825],{},[157,1409,1410],{"encoding":159},"rx = xr",[114,1412,1414,1436],{"className":1413,"ariaHidden":143},[164],[114,1415,1417,1421,1424,1427,1430,1433],{"className":1416},[168],[114,1418],{"className":1419,"style":1420},[172],"height:0.4306em;",[114,1422,825],{"className":1423,"style":856},[181,182],[114,1425,835],{"className":1426},[181,182],[114,1428],{"className":1429,"style":250},[191],[114,1431,1403],{"className":1432},[254],[114,1434],{"className":1435,"style":250},[191],[114,1437,1439,1442,1445],{"className":1438},[168],[114,1440],{"className":1441,"style":1420},[172],[114,1443,835],{"className":1444},[181,182],[114,1446,825],{"className":1447,"style":856},[181,182],"，此时左理想、右理想、理想是等价概念。",[106,1450,1452],{"id":1451},"_32-补充案例分析例-321-详解","3.2 补充案例分析（例 3.2.1 详解）",[19,1454,1455],{},"这是一个经典的非交换环例子，老师课堂上略过了一部分，这里补充完整。",[19,1457,1458,1461,1462,1522,1523,1562,1563,1591,1592,1595,1596],{},[51,1459,1460],{},"例 3.2.1","：全矩阵环 ",[114,1463,1465,1491],{"className":1464},[117],[114,1466,1468],{"className":1467},[121],[123,1469,1470],{"xmlns":125},[127,1471,1472,1488],{},[130,1473,1474,1477,1479,1482,1484,1486],{},[138,1475,1476],{},"M",[133,1478,136],{"stretchy":135},[138,1480,1481],{},"n",[133,1483,144],{"separator":143},[138,1485,140],{"mathvariant":555},[133,1487,155],{"stretchy":135},[157,1489,1490],{"encoding":159},"M(n, \\mathbb{R})",[114,1492,1494],{"className":1493,"ariaHidden":143},[164],[114,1495,1497,1500,1504,1507,1510,1513,1516,1519],{"className":1496},[168],[114,1498],{"className":1499,"style":173},[172],[114,1501,1476],{"className":1502,"style":1503},[181,182],"margin-right:0.109em;",[114,1505,136],{"className":1506},[177],[114,1508,1481],{"className":1509},[181,182],[114,1511,144],{"className":1512},[187],[114,1514],{"className":1515,"style":192},[191],[114,1517,140],{"className":1518},[181,576],[114,1520,155],{"className":1521},[208]," 只有平凡理想（即 ",[114,1524,1526,1544],{"className":1525},[117],[114,1527,1529],{"className":1528},[121],[123,1530,1531],{"xmlns":125},[127,1532,1533,1541],{},[130,1534,1535,1537,1539],{},[133,1536,626],{"stretchy":135},[550,1538,629],{},[133,1540,632],{"stretchy":135},[157,1542,1543],{"encoding":159},"\\{0\\}",[114,1545,1547],{"className":1546,"ariaHidden":143},[164],[114,1548,1550,1553,1556,1559],{"className":1549},[168],[114,1551],{"className":1552,"style":173},[172],[114,1554,626],{"className":1555},[177],[114,1557,629],{"className":1558},[181],[114,1560,632],{"className":1561},[208]," 和 ",[114,1564,1566,1579],{"className":1565},[117],[114,1567,1569],{"className":1568},[121],[123,1570,1571],{"xmlns":125},[127,1572,1573,1577],{},[130,1574,1575],{},[138,1576,140],{},[157,1578,140],{"encoding":159},[114,1580,1582],{"className":1581,"ariaHidden":143},[164],[114,1583,1585,1588],{"className":1584},[168],[114,1586],{"className":1587,"style":264},[172],[114,1589,140],{"className":1590,"style":183},[181,182]," 本身）。这种只有平凡理想的环称为",[51,1593,1594],{},"单环 (Simple Ring)","。\n",[533,1597,1598],{},"但是，它拥有非平凡的左理想或右理想。",[19,1600,1601,1604,1605,363,1633,1689,1690,445],{},[51,1602,1603],{},"证明部分（单环性质）：","\n设 ",[114,1606,1608,1621],{"className":1607},[117],[114,1609,1611],{"className":1610},[121],[123,1612,1613],{"xmlns":125},[127,1614,1615,1619],{},[130,1616,1617],{},[138,1618,756],{},[157,1620,756],{"encoding":159},[114,1622,1624],{"className":1623,"ariaHidden":143},[164],[114,1625,1627,1630],{"className":1626},[168],[114,1628],{"className":1629,"style":264},[172],[114,1631,756],{"className":1632,"style":771},[181,182],[114,1634,1636,1659],{"className":1635},[117],[114,1637,1639],{"className":1638},[121],[123,1640,1641],{"xmlns":125},[127,1642,1643,1657],{},[130,1644,1645,1647,1649,1651,1653,1655],{},[138,1646,1476],{},[133,1648,136],{"stretchy":135},[138,1650,1481],{},[133,1652,144],{"separator":143},[138,1654,140],{"mathvariant":555},[133,1656,155],{"stretchy":135},[157,1658,1490],{"encoding":159},[114,1660,1662],{"className":1661,"ariaHidden":143},[164],[114,1663,1665,1668,1671,1674,1677,1680,1683,1686],{"className":1664},[168],[114,1666],{"className":1667,"style":173},[172],[114,1669,1476],{"className":1670,"style":1503},[181,182],[114,1672,136],{"className":1673},[177],[114,1675,1481],{"className":1676},[181,182],[114,1678,144],{"className":1679},[187],[114,1681],{"className":1682,"style":192},[191],[114,1684,140],{"className":1685},[181,576],[114,1687,155],{"className":1688},[208]," 的一个非零理想。我们要证明 ",[114,1691,1693,1721],{"className":1692},[117],[114,1694,1696],{"className":1695},[121],[123,1697,1698],{"xmlns":125},[127,1699,1700,1718],{},[130,1701,1702,1704,1706,1708,1710,1712,1714,1716],{},[138,1703,756],{},[133,1705,1403],{},[138,1707,1476],{},[133,1709,136],{"stretchy":135},[138,1711,1481],{},[133,1713,144],{"separator":143},[138,1715,140],{"mathvariant":555},[133,1717,155],{"stretchy":135},[157,1719,1720],{"encoding":159},"I = M(n, \\mathbb{R})",[114,1722,1724,1742],{"className":1723,"ariaHidden":143},[164],[114,1725,1727,1730,1733,1736,1739],{"className":1726},[168],[114,1728],{"className":1729,"style":264},[172],[114,1731,756],{"className":1732,"style":771},[181,182],[114,1734],{"className":1735,"style":250},[191],[114,1737,1403],{"className":1738},[254],[114,1740],{"className":1741,"style":250},[191],[114,1743,1745,1748,1751,1754,1757,1760,1763,1766],{"className":1744},[168],[114,1746],{"className":1747,"style":173},[172],[114,1749,1476],{"className":1750,"style":1503},[181,182],[114,1752,136],{"className":1753},[177],[114,1755,1481],{"className":1756},[181,182],[114,1758,144],{"className":1759},[187],[114,1761],{"className":1762,"style":192},[191],[114,1764,140],{"className":1765},[181,576],[114,1767,155],{"className":1768},[208],[803,1770,1771,2065,2379,3634,4180,4881,5176],{},[450,1772,1773,1774,1878,1879,1932,1933,445],{},"因为 ",[114,1775,1777,1801],{"className":1776},[117],[114,1778,1780],{"className":1779},[121],[123,1781,1782],{"xmlns":125},[127,1783,1784,1798],{},[130,1785,1786,1788,1792,1794,1796],{},[138,1787,756],{},[133,1789,1791],{"mathvariant":1790},"normal","≠",[133,1793,626],{"stretchy":135},[550,1795,629],{},[133,1797,632],{"stretchy":135},[157,1799,1800],{"encoding":159},"I \\neq \\{0\\}",[114,1802,1804,1863],{"className":1803,"ariaHidden":143},[164],[114,1805,1807,1811,1814,1817,1860],{"className":1806},[168],[114,1808],{"className":1809,"style":1810},[172],"height:0.8889em;vertical-align:-0.1944em;",[114,1812,756],{"className":1813,"style":771},[181,182],[114,1815],{"className":1816,"style":250},[191],[114,1818,1820,1853,1857],{"className":1819},[254],[114,1821,1823],{"className":1822},[254],[114,1824,1827],{"className":1825},[181,1826],"vbox",[114,1828,1831],{"className":1829},[1830],"thinbox",[114,1832,1835,1838,1849],{"className":1833},[1834],"rlap",[114,1836],{"className":1837,"style":1810},[172],[114,1839,1842],{"className":1840},[1841],"inner",[114,1843,1845],{"className":1844},[181],[114,1846,1848],{"className":1847},[254],"",[114,1850],{"className":1851},[1852],"fix",[114,1854],{"className":1855},[191,1856],"nobreak",[114,1858,1403],{"className":1859},[254],[114,1861],{"className":1862,"style":250},[191],[114,1864,1866,1869,1872,1875],{"className":1865},[168],[114,1867],{"className":1868,"style":173},[172],[114,1870,626],{"className":1871},[177],[114,1873,629],{"className":1874},[181],[114,1876,632],{"className":1877},[208],"，所以存在非零矩阵 ",[114,1880,1882,1901],{"className":1881},[117],[114,1883,1885],{"className":1884},[121],[123,1886,1887],{"xmlns":125},[127,1888,1889,1898],{},[130,1890,1891,1894,1896],{},[138,1892,1893],{},"A",[133,1895,828],{},[138,1897,756],{},[157,1899,1900],{"encoding":159},"A \\in I",[114,1902,1904,1923],{"className":1903,"ariaHidden":143},[164],[114,1905,1907,1911,1914,1917,1920],{"className":1906},[168],[114,1908],{"className":1909,"style":1910},[172],"height:0.7224em;vertical-align:-0.0391em;",[114,1912,1893],{"className":1913},[181,182],[114,1915],{"className":1916,"style":250},[191],[114,1918,828],{"className":1919},[254],[114,1921],{"className":1922,"style":250},[191],[114,1924,1926,1929],{"className":1925},[168],[114,1927],{"className":1928,"style":264},[172],[114,1930,756],{"className":1931,"style":771},[181,182],"。记 ",[114,1934,1936,1970],{"className":1935},[117],[114,1937,1939],{"className":1938},[121],[123,1940,1941],{"xmlns":125},[127,1942,1943,1967],{},[130,1944,1945,1947,1949,1951,1965],{},[138,1946,1893],{},[133,1948,1403],{},[133,1950,136],{"stretchy":135},[1952,1953,1954,1957],"msub",{},[138,1955,1956],{},"a",[130,1958,1959,1962],{},[138,1960,1961],{},"i",[138,1963,1964],{},"j",[133,1966,155],{"stretchy":135},[157,1968,1969],{"encoding":159},"A = (a_{ij})",[114,1971,1973,1991],{"className":1972,"ariaHidden":143},[164],[114,1974,1976,1979,1982,1985,1988],{"className":1975},[168],[114,1977],{"className":1978,"style":264},[172],[114,1980,1893],{"className":1981},[181,182],[114,1983],{"className":1984,"style":250},[191],[114,1986,1403],{"className":1987},[254],[114,1989],{"className":1990,"style":250},[191],[114,1992,1994,1998,2001,2062],{"className":1993},[168],[114,1995],{"className":1996,"style":1997},[172],"height:1.0361em;vertical-align:-0.2861em;",[114,1999,136],{"className":2000},[177],[114,2002,2004,2007],{"className":2003},[181],[114,2005,1956],{"className":2006},[181,182],[114,2008,2011],{"className":2009},[2010],"msupsub",[114,2012,2016,2053],{"className":2013},[2014,2015],"vlist-t","vlist-t2",[114,2017,2020,2048],{"className":2018},[2019],"vlist-r",[114,2021,2025],{"className":2022,"style":2024},[2023],"vlist","height:0.3117em;",[114,2026,2028,2033],{"style":2027},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[114,2029],{"className":2030,"style":2032},[2031],"pstrut","height:2.7em;",[114,2034,2040],{"className":2035},[2036,2037,2038,2039],"sizing","reset-size6","size3","mtight",[114,2041,2043],{"className":2042},[181,2039],[114,2044,2047],{"className":2045,"style":2046},[181,182,2039],"margin-right:0.0572em;","ij",[114,2049,2052],{"className":2050},[2051],"vlist-s","​",[114,2054,2056],{"className":2055},[2019],[114,2057,2060],{"className":2058,"style":2059},[2023],"height:0.2861em;",[114,2061],{},[114,2063,155],{"className":2064},[208],[450,2066,2067,2068,2153,2154,445],{},"既然 ",[114,2069,2071,2089],{"className":2070},[117],[114,2072,2074],{"className":2073},[121],[123,2075,2076],{"xmlns":125},[127,2077,2078,2086],{},[130,2079,2080,2082,2084],{},[138,2081,1893],{},[133,2083,1791],{"mathvariant":1790},[550,2085,629],{},[157,2087,2088],{"encoding":159},"A \\neq 0",[114,2090,2092,2143],{"className":2091,"ariaHidden":143},[164],[114,2093,2095,2098,2101,2104,2140],{"className":2094},[168],[114,2096],{"className":2097,"style":1810},[172],[114,2099,1893],{"className":2100},[181,182],[114,2102],{"className":2103,"style":250},[191],[114,2105,2107,2134,2137],{"className":2106},[254],[114,2108,2110],{"className":2109},[254],[114,2111,2113],{"className":2112},[181,1826],[114,2114,2116],{"className":2115},[1830],[114,2117,2119,2122,2131],{"className":2118},[1834],[114,2120],{"className":2121,"style":1810},[172],[114,2123,2125],{"className":2124},[1841],[114,2126,2128],{"className":2127},[181],[114,2129,1848],{"className":2130},[254],[114,2132],{"className":2133},[1852],[114,2135],{"className":2136},[191,1856],[114,2138,1403],{"className":2139},[254],[114,2141],{"className":2142,"style":250},[191],[114,2144,2146,2150],{"className":2145},[168],[114,2147],{"className":2148,"style":2149},[172],"height:0.6444em;",[114,2151,629],{"className":2152},[181],"，至少存在一个元素 ",[114,2155,2157,2191],{"className":2156},[117],[114,2158,2160],{"className":2159},[121],[123,2161,2162],{"xmlns":125},[127,2163,2164,2188],{},[130,2165,2166,2184,2186],{},[1952,2167,2168,2170],{},[138,2169,1956],{},[130,2171,2172,2178],{},[1952,2173,2174,2176],{},[138,2175,1961],{},[550,2177,629],{},[1952,2179,2180,2182],{},[138,2181,1964],{},[550,2183,629],{},[133,2185,1791],{"mathvariant":1790},[550,2187,629],{},[157,2189,2190],{"encoding":159},"a_{i_0 j_0} \\neq 0",[114,2192,2194,2370],{"className":2193,"ariaHidden":143},[164],[114,2195,2197,2201,2328,2331,2367],{"className":2196},[168],[114,2198],{"className":2199,"style":2200},[172],"height:0.9805em;vertical-align:-0.2861em;",[114,2202,2204,2207],{"className":2203},[181],[114,2205,1956],{"className":2206},[181,182],[114,2208,2210],{"className":2209},[2010],[114,2211,2213,2320],{"className":2212},[2014,2015],[114,2214,2216,2317],{"className":2215},[2019],[114,2217,2219],{"className":2218,"style":2024},[2023],[114,2220,2221,2224],{"style":2027},[114,2222],{"className":2223,"style":2032},[2031],[114,2225,2227],{"className":2226},[2036,2037,2038,2039],[114,2228,2230,2276],{"className":2229},[181,2039],[114,2231,2233,2236],{"className":2232},[181,2039],[114,2234,1961],{"className":2235},[181,182,2039],[114,2237,2239],{"className":2238},[2010],[114,2240,2242,2267],{"className":2241},[2014,2015],[114,2243,2245,2264],{"className":2244},[2019],[114,2246,2249],{"className":2247,"style":2248},[2023],"height:0.3173em;",[114,2250,2252,2256],{"style":2251},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[114,2253],{"className":2254,"style":2255},[2031],"height:2.5em;",[114,2257,2261],{"className":2258},[2036,2259,2260,2039],"reset-size3","size1",[114,2262,629],{"className":2263},[181,2039],[114,2265,2052],{"className":2266},[2051],[114,2268,2270],{"className":2269},[2019],[114,2271,2274],{"className":2272,"style":2273},[2023],"height:0.143em;",[114,2275],{},[114,2277,2279,2282],{"className":2278},[181,2039],[114,2280,1964],{"className":2281,"style":2046},[181,182,2039],[114,2283,2285],{"className":2284},[2010],[114,2286,2288,2309],{"className":2287},[2014,2015],[114,2289,2291,2306],{"className":2290},[2019],[114,2292,2294],{"className":2293,"style":2248},[2023],[114,2295,2297,2300],{"style":2296},"top:-2.357em;margin-left:-0.0572em;margin-right:0.0714em;",[114,2298],{"className":2299,"style":2255},[2031],[114,2301,2303],{"className":2302},[2036,2259,2260,2039],[114,2304,629],{"className":2305},[181,2039],[114,2307,2052],{"className":2308},[2051],[114,2310,2312],{"className":2311},[2019],[114,2313,2315],{"className":2314,"style":2273},[2023],[114,2316],{},[114,2318,2052],{"className":2319},[2051],[114,2321,2323],{"className":2322},[2019],[114,2324,2326],{"className":2325,"style":2059},[2023],[114,2327],{},[114,2329],{"className":2330,"style":250},[191],[114,2332,2334,2361,2364],{"className":2333},[254],[114,2335,2337],{"className":2336},[254],[114,2338,2340],{"className":2339},[181,1826],[114,2341,2343],{"className":2342},[1830],[114,2344,2346,2349,2358],{"className":2345},[1834],[114,2347],{"className":2348,"style":1810},[172],[114,2350,2352],{"className":2351},[1841],[114,2353,2355],{"className":2354},[181],[114,2356,1848],{"className":2357},[254],[114,2359],{"className":2360},[1852],[114,2362],{"className":2363},[191,1856],[114,2365,1403],{"className":2366},[254],[114,2368],{"className":2369,"style":250},[191],[114,2371,2373,2376],{"className":2372},[168],[114,2374],{"className":2375,"style":2149},[172],[114,2377,629],{"className":2378},[181],[450,2380,2381,2382,2466,2467,2496,2497,2525,2526,2750,2751,2779,2780,2853,2854,2883,2884,2956,2957,2985,2986,3038,3039,3202,3203],{},"利用矩阵单位 ",[114,2383,2385,2409],{"className":2384},[117],[114,2386,2388],{"className":2387},[121],[123,2389,2390],{"xmlns":125},[127,2391,2392,2406],{},[130,2393,2394],{},[1952,2395,2396,2399],{},[138,2397,2398],{},"E",[130,2400,2401,2403],{},[138,2402,19],{},[138,2404,2405],{},"q",[157,2407,2408],{"encoding":159},"E_{pq}",[114,2410,2412],{"className":2411,"ariaHidden":143},[164],[114,2413,2415,2419],{"className":2414},[168],[114,2416],{"className":2417,"style":2418},[172],"height:0.9694em;vertical-align:-0.2861em;",[114,2420,2422,2425],{"className":2421},[181],[114,2423,2398],{"className":2424,"style":246},[181,182],[114,2426,2428],{"className":2427},[2010],[114,2429,2431,2458],{"className":2430},[2014,2015],[114,2432,2434,2455],{"className":2433},[2019],[114,2435,2438],{"className":2436,"style":2437},[2023],"height:0.1514em;",[114,2439,2441,2444],{"style":2440},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[114,2442],{"className":2443,"style":2032},[2031],[114,2445,2447],{"className":2446},[2036,2037,2038,2039],[114,2448,2450],{"className":2449},[181,2039],[114,2451,2454],{"className":2452,"style":2453},[181,182,2039],"margin-right:0.0359em;","pq",[114,2456,2052],{"className":2457},[2051],[114,2459,2461],{"className":2460},[2019],[114,2462,2464],{"className":2463,"style":2059},[2023],[114,2465],{},"（第 ",[114,2468,2470,2483],{"className":2469},[117],[114,2471,2473],{"className":2472},[121],[123,2474,2475],{"xmlns":125},[127,2476,2477,2481],{},[130,2478,2479],{},[138,2480,19],{},[157,2482,19],{"encoding":159},[114,2484,2486],{"className":2485,"ariaHidden":143},[164],[114,2487,2489,2493],{"className":2488},[168],[114,2490],{"className":2491,"style":2492},[172],"height:0.625em;vertical-align:-0.1944em;",[114,2494,19],{"className":2495},[181,182]," 行第 ",[114,2498,2500,2513],{"className":2499},[117],[114,2501,2503],{"className":2502},[121],[123,2504,2505],{"xmlns":125},[127,2506,2507,2511],{},[130,2508,2509],{},[138,2510,2405],{},[157,2512,2405],{"encoding":159},[114,2514,2516],{"className":2515,"ariaHidden":143},[164],[114,2517,2519,2522],{"className":2518},[168],[114,2520],{"className":2521,"style":2492},[172],[114,2523,2405],{"className":2524,"style":2453},[181,182]," 列为 1，其余为 0 的矩阵）。我们构造运算：\n",[114,2527,2529,2571],{"className":2528},[117],[114,2530,2532],{"className":2531},[121],[123,2533,2534],{"xmlns":125},[127,2535,2536,2568],{},[130,2537,2538,2552,2554],{},[1952,2539,2540,2542],{},[138,2541,2398],{},[130,2543,2544,2546],{},[138,2545,1961],{},[1952,2547,2548,2550],{},[138,2549,1961],{},[550,2551,629],{},[138,2553,1893],{},[1952,2555,2556,2558],{},[138,2557,2398],{},[130,2559,2560,2566],{},[1952,2561,2562,2564],{},[138,2563,1964],{},[550,2565,629],{},[138,2567,1964],{},[157,2569,2570],{"encoding":159},"E_{i i_0} A E_{j_0 j}",[114,2572,2574],{"className":2573,"ariaHidden":143},[164],[114,2575,2577,2580,2664,2667],{"className":2576},[168],[114,2578],{"className":2579,"style":2418},[172],[114,2581,2583,2586],{"className":2582},[181],[114,2584,2398],{"className":2585,"style":246},[181,182],[114,2587,2589],{"className":2588},[2010],[114,2590,2592,2655],{"className":2591},[2014,2015],[114,2593,2595,2652],{"className":2594},[2019],[114,2596,2598],{"className":2597,"style":2024},[2023],[114,2599,2600,2603],{"style":2440},[114,2601],{"className":2602,"style":2032},[2031],[114,2604,2606],{"className":2605},[2036,2037,2038,2039],[114,2607,2609,2612],{"className":2608},[181,2039],[114,2610,1961],{"className":2611},[181,182,2039],[114,2613,2615,2618],{"className":2614},[181,2039],[114,2616,1961],{"className":2617},[181,182,2039],[114,2619,2621],{"className":2620},[2010],[114,2622,2624,2644],{"className":2623},[2014,2015],[114,2625,2627,2641],{"className":2626},[2019],[114,2628,2630],{"className":2629,"style":2248},[2023],[114,2631,2632,2635],{"style":2251},[114,2633],{"className":2634,"style":2255},[2031],[114,2636,2638],{"className":2637},[2036,2259,2260,2039],[114,2639,629],{"className":2640},[181,2039],[114,2642,2052],{"className":2643},[2051],[114,2645,2647],{"className":2646},[2019],[114,2648,2650],{"className":2649,"style":2273},[2023],[114,2651],{},[114,2653,2052],{"className":2654},[2051],[114,2656,2658],{"className":2657},[2019],[114,2659,2662],{"className":2660,"style":2661},[2023],"height:0.2501em;",[114,2663],{},[114,2665,1893],{"className":2666},[181,182],[114,2668,2670,2673],{"className":2669},[181],[114,2671,2398],{"className":2672,"style":246},[181,182],[114,2674,2676],{"className":2675},[2010],[114,2677,2679,2742],{"className":2678},[2014,2015],[114,2680,2682,2739],{"className":2681},[2019],[114,2683,2685],{"className":2684,"style":2024},[2023],[114,2686,2687,2690],{"style":2440},[114,2688],{"className":2689,"style":2032},[2031],[114,2691,2693],{"className":2692},[2036,2037,2038,2039],[114,2694,2696,2736],{"className":2695},[181,2039],[114,2697,2699,2702],{"className":2698},[181,2039],[114,2700,1964],{"className":2701,"style":2046},[181,182,2039],[114,2703,2705],{"className":2704},[2010],[114,2706,2708,2728],{"className":2707},[2014,2015],[114,2709,2711,2725],{"className":2710},[2019],[114,2712,2714],{"className":2713,"style":2248},[2023],[114,2715,2716,2719],{"style":2296},[114,2717],{"className":2718,"style":2255},[2031],[114,2720,2722],{"className":2721},[2036,2259,2260,2039],[114,2723,629],{"className":2724},[181,2039],[114,2726,2052],{"className":2727},[2051],[114,2729,2731],{"className":2730},[2019],[114,2732,2734],{"className":2733,"style":2273},[2023],[114,2735],{},[114,2737,1964],{"className":2738,"style":2046},[181,182,2039],[114,2740,2052],{"className":2741},[2051],[114,2743,2745],{"className":2744},[2019],[114,2746,2748],{"className":2747,"style":2059},[2023],[114,2749],{},"\n这个运算的效果是将 ",[114,2752,2754,2767],{"className":2753},[117],[114,2755,2757],{"className":2756},[121],[123,2758,2759],{"xmlns":125},[127,2760,2761,2765],{},[130,2762,2763],{},[138,2764,1893],{},[157,2766,1893],{"encoding":159},[114,2768,2770],{"className":2769,"ariaHidden":143},[164],[114,2771,2773,2776],{"className":2772},[168],[114,2774],{"className":2775,"style":264},[172],[114,2777,1893],{"className":2778},[181,182]," 的第 ",[114,2781,2783,2801],{"className":2782},[117],[114,2784,2786],{"className":2785},[121],[123,2787,2788],{"xmlns":125},[127,2789,2790,2798],{},[130,2791,2792],{},[1952,2793,2794,2796],{},[138,2795,1961],{},[550,2797,629],{},[157,2799,2800],{"encoding":159},"i_0",[114,2802,2804],{"className":2803,"ariaHidden":143},[164],[114,2805,2807,2811],{"className":2806},[168],[114,2808],{"className":2809,"style":2810},[172],"height:0.8095em;vertical-align:-0.15em;",[114,2812,2814,2817],{"className":2813},[181],[114,2815,1961],{"className":2816},[181,182],[114,2818,2820],{"className":2819},[2010],[114,2821,2823,2844],{"className":2822},[2014,2015],[114,2824,2826,2841],{"className":2825},[2019],[114,2827,2830],{"className":2828,"style":2829},[2023],"height:0.3011em;",[114,2831,2832,2835],{"style":2027},[114,2833],{"className":2834,"style":2032},[2031],[114,2836,2838],{"className":2837},[2036,2037,2038,2039],[114,2839,629],{"className":2840},[181,2039],[114,2842,2052],{"className":2843},[2051],[114,2845,2847],{"className":2846},[2019],[114,2848,2851],{"className":2849,"style":2850},[2023],"height:0.15em;",[114,2852],{}," 行移到第 ",[114,2855,2857,2870],{"className":2856},[117],[114,2858,2860],{"className":2859},[121],[123,2861,2862],{"xmlns":125},[127,2863,2864,2868],{},[130,2865,2866],{},[138,2867,1961],{},[157,2869,1961],{"encoding":159},[114,2871,2873],{"className":2872,"ariaHidden":143},[164],[114,2874,2876,2880],{"className":2875},[168],[114,2877],{"className":2878,"style":2879},[172],"height:0.6595em;",[114,2881,1961],{"className":2882},[181,182]," 行，第 ",[114,2885,2887,2905],{"className":2886},[117],[114,2888,2890],{"className":2889},[121],[123,2891,2892],{"xmlns":125},[127,2893,2894,2902],{},[130,2895,2896],{},[1952,2897,2898,2900],{},[138,2899,1964],{},[550,2901,629],{},[157,2903,2904],{"encoding":159},"j_0",[114,2906,2908],{"className":2907,"ariaHidden":143},[164],[114,2909,2911,2915],{"className":2910},[168],[114,2912],{"className":2913,"style":2914},[172],"height:0.854em;vertical-align:-0.1944em;",[114,2916,2918,2921],{"className":2917},[181],[114,2919,1964],{"className":2920,"style":2046},[181,182],[114,2922,2924],{"className":2923},[2010],[114,2925,2927,2948],{"className":2926},[2014,2015],[114,2928,2930,2945],{"className":2929},[2019],[114,2931,2933],{"className":2932,"style":2829},[2023],[114,2934,2936,2939],{"style":2935},"top:-2.55em;margin-left:-0.0572em;margin-right:0.05em;",[114,2937],{"className":2938,"style":2032},[2031],[114,2940,2942],{"className":2941},[2036,2037,2038,2039],[114,2943,629],{"className":2944},[181,2039],[114,2946,2052],{"className":2947},[2051],[114,2949,2951],{"className":2950},[2019],[114,2952,2954],{"className":2953,"style":2850},[2023],[114,2955],{}," 列移到第 ",[114,2958,2960,2973],{"className":2959},[117],[114,2961,2963],{"className":2962},[121],[123,2964,2965],{"xmlns":125},[127,2966,2967,2971],{},[130,2968,2969],{},[138,2970,1964],{},[157,2972,1964],{"encoding":159},[114,2974,2976],{"className":2975,"ariaHidden":143},[164],[114,2977,2979,2982],{"className":2978},[168],[114,2980],{"className":2981,"style":2914},[172],[114,2983,1964],{"className":2984,"style":2046},[181,182]," 列，并把其他元素“切除”。结果仅在 ",[114,2987,2989,3011],{"className":2988},[117],[114,2990,2992],{"className":2991},[121],[123,2993,2994],{"xmlns":125},[127,2995,2996,3008],{},[130,2997,2998,3000,3002,3004,3006],{},[133,2999,136],{"stretchy":135},[138,3001,1961],{},[133,3003,144],{"separator":143},[138,3005,1964],{},[133,3007,155],{"stretchy":135},[157,3009,3010],{"encoding":159},"(i, j)",[114,3012,3014],{"className":3013,"ariaHidden":143},[164],[114,3015,3017,3020,3023,3026,3029,3032,3035],{"className":3016},[168],[114,3018],{"className":3019,"style":173},[172],[114,3021,136],{"className":3022},[177],[114,3024,1961],{"className":3025},[181,182],[114,3027,144],{"className":3028},[187],[114,3030],{"className":3031,"style":192},[191],[114,3033,1964],{"className":3034,"style":2046},[181,182],[114,3036,155],{"className":3037},[208]," 位置留下 ",[114,3040,3042,3072],{"className":3041},[117],[114,3043,3045],{"className":3044},[121],[123,3046,3047],{"xmlns":125},[127,3048,3049,3069],{},[130,3050,3051],{},[1952,3052,3053,3055],{},[138,3054,1956],{},[130,3056,3057,3063],{},[1952,3058,3059,3061],{},[138,3060,1961],{},[550,3062,629],{},[1952,3064,3065,3067],{},[138,3066,1964],{},[550,3068,629],{},[157,3070,3071],{"encoding":159},"a_{i_0 j_0}",[114,3073,3075],{"className":3074,"ariaHidden":143},[164],[114,3076,3078,3082],{"className":3077},[168],[114,3079],{"className":3080,"style":3081},[172],"height:0.7167em;vertical-align:-0.2861em;",[114,3083,3085,3088],{"className":3084},[181],[114,3086,1956],{"className":3087},[181,182],[114,3089,3091],{"className":3090},[2010],[114,3092,3094,3194],{"className":3093},[2014,2015],[114,3095,3097,3191],{"className":3096},[2019],[114,3098,3100],{"className":3099,"style":2024},[2023],[114,3101,3102,3105],{"style":2027},[114,3103],{"className":3104,"style":2032},[2031],[114,3106,3108],{"className":3107},[2036,2037,2038,2039],[114,3109,3111,3151],{"className":3110},[181,2039],[114,3112,3114,3117],{"className":3113},[181,2039],[114,3115,1961],{"className":3116},[181,182,2039],[114,3118,3120],{"className":3119},[2010],[114,3121,3123,3143],{"className":3122},[2014,2015],[114,3124,3126,3140],{"className":3125},[2019],[114,3127,3129],{"className":3128,"style":2248},[2023],[114,3130,3131,3134],{"style":2251},[114,3132],{"className":3133,"style":2255},[2031],[114,3135,3137],{"className":3136},[2036,2259,2260,2039],[114,3138,629],{"className":3139},[181,2039],[114,3141,2052],{"className":3142},[2051],[114,3144,3146],{"className":3145},[2019],[114,3147,3149],{"className":3148,"style":2273},[2023],[114,3150],{},[114,3152,3154,3157],{"className":3153},[181,2039],[114,3155,1964],{"className":3156,"style":2046},[181,182,2039],[114,3158,3160],{"className":3159},[2010],[114,3161,3163,3183],{"className":3162},[2014,2015],[114,3164,3166,3180],{"className":3165},[2019],[114,3167,3169],{"className":3168,"style":2248},[2023],[114,3170,3171,3174],{"style":2296},[114,3172],{"className":3173,"style":2255},[2031],[114,3175,3177],{"className":3176},[2036,2259,2260,2039],[114,3178,629],{"className":3179},[181,2039],[114,3181,2052],{"className":3182},[2051],[114,3184,3186],{"className":3185},[2019],[114,3187,3189],{"className":3188,"style":2273},[2023],[114,3190],{},[114,3192,2052],{"className":3193},[2051],[114,3195,3197],{"className":3196},[2019],[114,3198,3200],{"className":3199,"style":2059},[2023],[114,3201],{},"。即：\n",[114,3204,3206,3278],{"className":3205},[117],[114,3207,3209],{"className":3208},[121],[123,3210,3211],{"xmlns":125},[127,3212,3213,3275],{},[130,3214,3215,3229,3231,3245,3247,3265],{},[1952,3216,3217,3219],{},[138,3218,2398],{},[130,3220,3221,3223],{},[138,3222,1961],{},[1952,3224,3225,3227],{},[138,3226,1961],{},[550,3228,629],{},[138,3230,1893],{},[1952,3232,3233,3235],{},[138,3234,2398],{},[130,3236,3237,3243],{},[1952,3238,3239,3241],{},[138,3240,1964],{},[550,3242,629],{},[138,3244,1964],{},[133,3246,1403],{},[1952,3248,3249,3251],{},[138,3250,1956],{},[130,3252,3253,3259],{},[1952,3254,3255,3257],{},[138,3256,1961],{},[550,3258,629],{},[1952,3260,3261,3263],{},[138,3262,1964],{},[550,3264,629],{},[1952,3266,3267,3269],{},[138,3268,2398],{},[130,3270,3271,3273],{},[138,3272,1961],{},[138,3274,1964],{},[157,3276,3277],{"encoding":159},"E_{i i_0} A E_{j_0 j} = a_{i_0 j_0} E_{ij}",[114,3279,3281,3465],{"className":3280,"ariaHidden":143},[164],[114,3282,3284,3287,3370,3373,3456,3459,3462],{"className":3283},[168],[114,3285],{"className":3286,"style":2418},[172],[114,3288,3290,3293],{"className":3289},[181],[114,3291,2398],{"className":3292,"style":246},[181,182],[114,3294,3296],{"className":3295},[2010],[114,3297,3299,3362],{"className":3298},[2014,2015],[114,3300,3302,3359],{"className":3301},[2019],[114,3303,3305],{"className":3304,"style":2024},[2023],[114,3306,3307,3310],{"style":2440},[114,3308],{"className":3309,"style":2032},[2031],[114,3311,3313],{"className":3312},[2036,2037,2038,2039],[114,3314,3316,3319],{"className":3315},[181,2039],[114,3317,1961],{"className":3318},[181,182,2039],[114,3320,3322,3325],{"className":3321},[181,2039],[114,3323,1961],{"className":3324},[181,182,2039],[114,3326,3328],{"className":3327},[2010],[114,3329,3331,3351],{"className":3330},[2014,2015],[114,3332,3334,3348],{"className":3333},[2019],[114,3335,3337],{"className":3336,"style":2248},[2023],[114,3338,3339,3342],{"style":2251},[114,3340],{"className":3341,"style":2255},[2031],[114,3343,3345],{"className":3344},[2036,2259,2260,2039],[114,3346,629],{"className":3347},[181,2039],[114,3349,2052],{"className":3350},[2051],[114,3352,3354],{"className":3353},[2019],[114,3355,3357],{"className":3356,"style":2273},[2023],[114,3358],{},[114,3360,2052],{"className":3361},[2051],[114,3363,3365],{"className":3364},[2019],[114,3366,3368],{"className":3367,"style":2661},[2023],[114,3369],{},[114,3371,1893],{"className":3372},[181,182],[114,3374,3376,3379],{"className":3375},[181],[114,3377,2398],{"className":3378,"style":246},[181,182],[114,3380,3382],{"className":3381},[2010],[114,3383,3385,3448],{"className":3384},[2014,2015],[114,3386,3388,3445],{"className":3387},[2019],[114,3389,3391],{"className":3390,"style":2024},[2023],[114,3392,3393,3396],{"style":2440},[114,3394],{"className":3395,"style":2032},[2031],[114,3397,3399],{"className":3398},[2036,2037,2038,2039],[114,3400,3402,3442],{"className":3401},[181,2039],[114,3403,3405,3408],{"className":3404},[181,2039],[114,3406,1964],{"className":3407,"style":2046},[181,182,2039],[114,3409,3411],{"className":3410},[2010],[114,3412,3414,3434],{"className":3413},[2014,2015],[114,3415,3417,3431],{"className":3416},[2019],[114,3418,3420],{"className":3419,"style":2248},[2023],[114,3421,3422,3425],{"style":2296},[114,3423],{"className":3424,"style":2255},[2031],[114,3426,3428],{"className":3427},[2036,2259,2260,2039],[114,3429,629],{"className":3430},[181,2039],[114,3432,2052],{"className":3433},[2051],[114,3435,3437],{"className":3436},[2019],[114,3438,3440],{"className":3439,"style":2273},[2023],[114,3441],{},[114,3443,1964],{"className":3444,"style":2046},[181,182,2039],[114,3446,2052],{"className":3447},[2051],[114,3449,3451],{"className":3450},[2019],[114,3452,3454],{"className":3453,"style":2059},[2023],[114,3455],{},[114,3457],{"className":3458,"style":250},[191],[114,3460,1403],{"className":3461},[254],[114,3463],{"className":3464,"style":250},[191],[114,3466,3468,3471,3591],{"className":3467},[168],[114,3469],{"className":3470,"style":2418},[172],[114,3472,3474,3477],{"className":3473},[181],[114,3475,1956],{"className":3476},[181,182],[114,3478,3480],{"className":3479},[2010],[114,3481,3483,3583],{"className":3482},[2014,2015],[114,3484,3486,3580],{"className":3485},[2019],[114,3487,3489],{"className":3488,"style":2024},[2023],[114,3490,3491,3494],{"style":2027},[114,3492],{"className":3493,"style":2032},[2031],[114,3495,3497],{"className":3496},[2036,2037,2038,2039],[114,3498,3500,3540],{"className":3499},[181,2039],[114,3501,3503,3506],{"className":3502},[181,2039],[114,3504,1961],{"className":3505},[181,182,2039],[114,3507,3509],{"className":3508},[2010],[114,3510,3512,3532],{"className":3511},[2014,2015],[114,3513,3515,3529],{"className":3514},[2019],[114,3516,3518],{"className":3517,"style":2248},[2023],[114,3519,3520,3523],{"style":2251},[114,3521],{"className":3522,"style":2255},[2031],[114,3524,3526],{"className":3525},[2036,2259,2260,2039],[114,3527,629],{"className":3528},[181,2039],[114,3530,2052],{"className":3531},[2051],[114,3533,3535],{"className":3534},[2019],[114,3536,3538],{"className":3537,"style":2273},[2023],[114,3539],{},[114,3541,3543,3546],{"className":3542},[181,2039],[114,3544,1964],{"className":3545,"style":2046},[181,182,2039],[114,3547,3549],{"className":3548},[2010],[114,3550,3552,3572],{"className":3551},[2014,2015],[114,3553,3555,3569],{"className":3554},[2019],[114,3556,3558],{"className":3557,"style":2248},[2023],[114,3559,3560,3563],{"style":2296},[114,3561],{"className":3562,"style":2255},[2031],[114,3564,3566],{"className":3565},[2036,2259,2260,2039],[114,3567,629],{"className":3568},[181,2039],[114,3570,2052],{"className":3571},[2051],[114,3573,3575],{"className":3574},[2019],[114,3576,3578],{"className":3577,"style":2273},[2023],[114,3579],{},[114,3581,2052],{"className":3582},[2051],[114,3584,3586],{"className":3585},[2019],[114,3587,3589],{"className":3588,"style":2059},[2023],[114,3590],{},[114,3592,3594,3597],{"className":3593},[181],[114,3595,2398],{"className":3596,"style":246},[181,182],[114,3598,3600],{"className":3599},[2010],[114,3601,3603,3626],{"className":3602},[2014,2015],[114,3604,3606,3623],{"className":3605},[2019],[114,3607,3609],{"className":3608,"style":2024},[2023],[114,3610,3611,3614],{"style":2440},[114,3612],{"className":3613,"style":2032},[2031],[114,3615,3617],{"className":3616},[2036,2037,2038,2039],[114,3618,3620],{"className":3619},[181,2039],[114,3621,2047],{"className":3622,"style":2046},[181,182,2039],[114,3624,2052],{"className":3625},[2051],[114,3627,3629],{"className":3628},[2019],[114,3630,3632],{"className":3631,"style":2059},[2023],[114,3633],{},[450,3635,1773,3636,3664,3665,3913,3914,3942,3943,445],{},[114,3637,3639,3652],{"className":3638},[117],[114,3640,3642],{"className":3641},[121],[123,3643,3644],{"xmlns":125},[127,3645,3646,3650],{},[130,3647,3648],{},[138,3649,756],{},[157,3651,756],{"encoding":159},[114,3653,3655],{"className":3654,"ariaHidden":143},[164],[114,3656,3658,3661],{"className":3657},[168],[114,3659],{"className":3660,"style":264},[172],[114,3662,756],{"className":3663,"style":771},[181,182]," 是理想，且 ",[114,3666,3668,3714],{"className":3667},[117],[114,3669,3671],{"className":3670},[121],[123,3672,3673],{"xmlns":125},[127,3674,3675,3711],{},[130,3676,3677,3691,3693,3707,3709],{},[1952,3678,3679,3681],{},[138,3680,2398],{},[130,3682,3683,3685],{},[138,3684,1961],{},[1952,3686,3687,3689],{},[138,3688,1961],{},[550,3690,629],{},[133,3692,144],{"separator":143},[1952,3694,3695,3697],{},[138,3696,2398],{},[130,3698,3699,3705],{},[1952,3700,3701,3703],{},[138,3702,1964],{},[550,3704,629],{},[138,3706,1964],{},[133,3708,828],{},[138,3710,140],{},[157,3712,3713],{"encoding":159},"E_{i i_0}, E_{j_0 j} \\in R",[114,3715,3717,3904],{"className":3716,"ariaHidden":143},[164],[114,3718,3720,3723,3806,3809,3812,3895,3898,3901],{"className":3719},[168],[114,3721],{"className":3722,"style":2418},[172],[114,3724,3726,3729],{"className":3725},[181],[114,3727,2398],{"className":3728,"style":246},[181,182],[114,3730,3732],{"className":3731},[2010],[114,3733,3735,3798],{"className":3734},[2014,2015],[114,3736,3738,3795],{"className":3737},[2019],[114,3739,3741],{"className":3740,"style":2024},[2023],[114,3742,3743,3746],{"style":2440},[114,3744],{"className":3745,"style":2032},[2031],[114,3747,3749],{"className":3748},[2036,2037,2038,2039],[114,3750,3752,3755],{"className":3751},[181,2039],[114,3753,1961],{"className":3754},[181,182,2039],[114,3756,3758,3761],{"className":3757},[181,2039],[114,3759,1961],{"className":3760},[181,182,2039],[114,3762,3764],{"className":3763},[2010],[114,3765,3767,3787],{"className":3766},[2014,2015],[114,3768,3770,3784],{"className":3769},[2019],[114,3771,3773],{"className":3772,"style":2248},[2023],[114,3774,3775,3778],{"style":2251},[114,3776],{"className":3777,"style":2255},[2031],[114,3779,3781],{"className":3780},[2036,2259,2260,2039],[114,3782,629],{"className":3783},[181,2039],[114,3785,2052],{"className":3786},[2051],[114,3788,3790],{"className":3789},[2019],[114,3791,3793],{"className":3792,"style":2273},[2023],[114,3794],{},[114,3796,2052],{"className":3797},[2051],[114,3799,3801],{"className":3800},[2019],[114,3802,3804],{"className":3803,"style":2661},[2023],[114,3805],{},[114,3807,144],{"className":3808},[187],[114,3810],{"className":3811,"style":192},[191],[114,3813,3815,3818],{"className":3814},[181],[114,3816,2398],{"className":3817,"style":246},[181,182],[114,3819,3821],{"className":3820},[2010],[114,3822,3824,3887],{"className":3823},[2014,2015],[114,3825,3827,3884],{"className":3826},[2019],[114,3828,3830],{"className":3829,"style":2024},[2023],[114,3831,3832,3835],{"style":2440},[114,3833],{"className":3834,"style":2032},[2031],[114,3836,3838],{"className":3837},[2036,2037,2038,2039],[114,3839,3841,3881],{"className":3840},[181,2039],[114,3842,3844,3847],{"className":3843},[181,2039],[114,3845,1964],{"className":3846,"style":2046},[181,182,2039],[114,3848,3850],{"className":3849},[2010],[114,3851,3853,3873],{"className":3852},[2014,2015],[114,3854,3856,3870],{"className":3855},[2019],[114,3857,3859],{"className":3858,"style":2248},[2023],[114,3860,3861,3864],{"style":2296},[114,3862],{"className":3863,"style":2255},[2031],[114,3865,3867],{"className":3866},[2036,2259,2260,2039],[114,3868,629],{"className":3869},[181,2039],[114,3871,2052],{"className":3872},[2051],[114,3874,3876],{"className":3875},[2019],[114,3877,3879],{"className":3878,"style":2273},[2023],[114,3880],{},[114,3882,1964],{"className":3883,"style":2046},[181,182,2039],[114,3885,2052],{"className":3886},[2051],[114,3888,3890],{"className":3889},[2019],[114,3891,3893],{"className":3892,"style":2059},[2023],[114,3894],{},[114,3896],{"className":3897,"style":250},[191],[114,3899,828],{"className":3900},[254],[114,3902],{"className":3903,"style":250},[191],[114,3905,3907,3910],{"className":3906},[168],[114,3908],{"className":3909,"style":264},[172],[114,3911,140],{"className":3912,"style":183},[181,182],"，根据吸收律，上式结果必须在 ",[114,3915,3917,3930],{"className":3916},[117],[114,3918,3920],{"className":3919},[121],[123,3921,3922],{"xmlns":125},[127,3923,3924,3928],{},[130,3925,3926],{},[138,3927,756],{},[157,3929,756],{"encoding":159},[114,3931,3933],{"className":3932,"ariaHidden":143},[164],[114,3934,3936,3939],{"className":3935},[168],[114,3937],{"className":3938,"style":264},[172],[114,3940,756],{"className":3941,"style":771},[181,182]," 中。\n即 ",[114,3944,3946,3990],{"className":3945},[117],[114,3947,3949],{"className":3948},[121],[123,3950,3951],{"xmlns":125},[127,3952,3953,3987],{},[130,3954,3955,3973,3983,3985],{},[1952,3956,3957,3959],{},[138,3958,1956],{},[130,3960,3961,3967],{},[1952,3962,3963,3965],{},[138,3964,1961],{},[550,3966,629],{},[1952,3968,3969,3971],{},[138,3970,1964],{},[550,3972,629],{},[1952,3974,3975,3977],{},[138,3976,2398],{},[130,3978,3979,3981],{},[138,3980,1961],{},[138,3982,1964],{},[133,3984,828],{},[138,3986,756],{},[157,3988,3989],{"encoding":159},"a_{i_0 j_0} E_{ij} \\in I",[114,3991,3993,4171],{"className":3992,"ariaHidden":143},[164],[114,3994,3996,3999,4119,4162,4165,4168],{"className":3995},[168],[114,3997],{"className":3998,"style":2418},[172],[114,4000,4002,4005],{"className":4001},[181],[114,4003,1956],{"className":4004},[181,182],[114,4006,4008],{"className":4007},[2010],[114,4009,4011,4111],{"className":4010},[2014,2015],[114,4012,4014,4108],{"className":4013},[2019],[114,4015,4017],{"className":4016,"style":2024},[2023],[114,4018,4019,4022],{"style":2027},[114,4020],{"className":4021,"style":2032},[2031],[114,4023,4025],{"className":4024},[2036,2037,2038,2039],[114,4026,4028,4068],{"className":4027},[181,2039],[114,4029,4031,4034],{"className":4030},[181,2039],[114,4032,1961],{"className":4033},[181,182,2039],[114,4035,4037],{"className":4036},[2010],[114,4038,4040,4060],{"className":4039},[2014,2015],[114,4041,4043,4057],{"className":4042},[2019],[114,4044,4046],{"className":4045,"style":2248},[2023],[114,4047,4048,4051],{"style":2251},[114,4049],{"className":4050,"style":2255},[2031],[114,4052,4054],{"className":4053},[2036,2259,2260,2039],[114,4055,629],{"className":4056},[181,2039],[114,4058,2052],{"className":4059},[2051],[114,4061,4063],{"className":4062},[2019],[114,4064,4066],{"className":4065,"style":2273},[2023],[114,4067],{},[114,4069,4071,4074],{"className":4070},[181,2039],[114,4072,1964],{"className":4073,"style":2046},[181,182,2039],[114,4075,4077],{"className":4076},[2010],[114,4078,4080,4100],{"className":4079},[2014,2015],[114,4081,4083,4097],{"className":4082},[2019],[114,4084,4086],{"className":4085,"style":2248},[2023],[114,4087,4088,4091],{"style":2296},[114,4089],{"className":4090,"style":2255},[2031],[114,4092,4094],{"className":4093},[2036,2259,2260,2039],[114,4095,629],{"className":4096},[181,2039],[114,4098,2052],{"className":4099},[2051],[114,4101,4103],{"className":4102},[2019],[114,4104,4106],{"className":4105,"style":2273},[2023],[114,4107],{},[114,4109,2052],{"className":4110},[2051],[114,4112,4114],{"className":4113},[2019],[114,4115,4117],{"className":4116,"style":2059},[2023],[114,4118],{},[114,4120,4122,4125],{"className":4121},[181],[114,4123,2398],{"className":4124,"style":246},[181,182],[114,4126,4128],{"className":4127},[2010],[114,4129,4131,4154],{"className":4130},[2014,2015],[114,4132,4134,4151],{"className":4133},[2019],[114,4135,4137],{"className":4136,"style":2024},[2023],[114,4138,4139,4142],{"style":2440},[114,4140],{"className":4141,"style":2032},[2031],[114,4143,4145],{"className":4144},[2036,2037,2038,2039],[114,4146,4148],{"className":4147},[181,2039],[114,4149,2047],{"className":4150,"style":2046},[181,182,2039],[114,4152,2052],{"className":4153},[2051],[114,4155,4157],{"className":4156},[2019],[114,4158,4160],{"className":4159,"style":2059},[2023],[114,4161],{},[114,4163],{"className":4164,"style":250},[191],[114,4166,828],{"className":4167},[254],[114,4169],{"className":4170,"style":250},[191],[114,4172,4174,4177],{"className":4173},[168],[114,4175],{"className":4176,"style":264},[172],[114,4178,756],{"className":4179,"style":771},[181,182],[450,4181,1773,4182,4343,4344],{},[114,4183,4185,4214],{"className":4184},[117],[114,4186,4188],{"className":4187},[121],[123,4189,4190],{"xmlns":125},[127,4191,4192,4212],{},[130,4193,4194],{},[1952,4195,4196,4198],{},[138,4197,1956],{},[130,4199,4200,4206],{},[1952,4201,4202,4204],{},[138,4203,1961],{},[550,4205,629],{},[1952,4207,4208,4210],{},[138,4209,1964],{},[550,4211,629],{},[157,4213,3071],{"encoding":159},[114,4215,4217],{"className":4216,"ariaHidden":143},[164],[114,4218,4220,4223],{"className":4219},[168],[114,4221],{"className":4222,"style":3081},[172],[114,4224,4226,4229],{"className":4225},[181],[114,4227,1956],{"className":4228},[181,182],[114,4230,4232],{"className":4231},[2010],[114,4233,4235,4335],{"className":4234},[2014,2015],[114,4236,4238,4332],{"className":4237},[2019],[114,4239,4241],{"className":4240,"style":2024},[2023],[114,4242,4243,4246],{"style":2027},[114,4244],{"className":4245,"style":2032},[2031],[114,4247,4249],{"className":4248},[2036,2037,2038,2039],[114,4250,4252,4292],{"className":4251},[181,2039],[114,4253,4255,4258],{"className":4254},[181,2039],[114,4256,1961],{"className":4257},[181,182,2039],[114,4259,4261],{"className":4260},[2010],[114,4262,4264,4284],{"className":4263},[2014,2015],[114,4265,4267,4281],{"className":4266},[2019],[114,4268,4270],{"className":4269,"style":2248},[2023],[114,4271,4272,4275],{"style":2251},[114,4273],{"className":4274,"style":2255},[2031],[114,4276,4278],{"className":4277},[2036,2259,2260,2039],[114,4279,629],{"className":4280},[181,2039],[114,4282,2052],{"className":4283},[2051],[114,4285,4287],{"className":4286},[2019],[114,4288,4290],{"className":4289,"style":2273},[2023],[114,4291],{},[114,4293,4295,4298],{"className":4294},[181,2039],[114,4296,1964],{"className":4297,"style":2046},[181,182,2039],[114,4299,4301],{"className":4300},[2010],[114,4302,4304,4324],{"className":4303},[2014,2015],[114,43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是非零实数，其逆元存在。利用理想性质再次乘以标量（或对角阵）：\n",[114,4345,4347,4431],{"className":4346},[117],[114,4348,4350],{"className":4349},[121],[123,4351,4352],{"xmlns":125},[127,4353,4354,4428],{},[130,4355,4356,4366,4368,4392,4394,4412,4422,4424,4426],{},[1952,4357,4358,4360],{},[138,4359,2398],{},[130,4361,4362,4364],{},[138,4363,1961],{},[138,4365,1964],{},[133,4367,1403],{},[4369,4370,4371,4374],"mfrac",{},[550,4372,4373],{},"1",[1952,4375,4376,4378],{},[138,4377,1956],{},[130,4379,4380,4386],{},[1952,4381,4382,4384],{},[138,4383,1961],{},[550,4385,629],{},[1952,4387,4388,4390],{},[138,4389,1964],{},[550,4391,629],{},[133,4393,136],{"stretchy":135},[1952,4395,4396,4398],{},[138,4397,1956],{},[130,4399,4400,4406],{},[1952,4401,4402,4404],{},[138,4403,1961],{},[550,4405,629],{},[1952,4407,4408,4410],{},[138,4409,1964],{},[550,4411,629],{},[1952,4413,4414,4416],{},[138,4415,2398],{},[130,4417,4418,4420],{},[138,4419,1961],{},[138,4421,1964],{},[133,4423,155],{"stretchy":135},[133,4425,828],{},[138,4427,756],{},[157,4429,4430],{"encoding":159},"E_{ij} = \\frac{1}{a_{i_0 j_0}} (a_{i_0 j_0} E_{ij}) \\in I",[114,4432,4434,4492,4872],{"className":4433,"ariaHidden":143},[164],[114,4435,4437,4440,4483,4486,4489],{"className":4436},[168],[114,4438],{"className":4439,"style":2418},[172],[114,4441,4443,4446],{"className":4442},[181],[114,4444,2398],{"className":4445,"style":246},[181,182],[114,4447,4449],{"className":4448},[2010],[114,4450,4452,4475],{"className":4451},[2014,2015],[114,4453,4455,4472],{"className":4454},[2019],[114,4456,4458],{"className":4457,"style":2024},[2023],[114,4459,4460,4463],{"style":2440},[114,4461],{"className":4462,"style":2032},[2031],[114,4464,4466],{"className":4465},[2036,2037,2038,2039],[114,4467,4469],{"className":4468},[181,2039],[114,4470,2047],{"className":4471,"style":2046},[181,182,2039],[114,4473,2052],{"className":4474},[2051],[114,4476,4478],{"className":4477},[2019],[114,4479,4481],{"className":4480,"style":2059},[2023],[114,4482],{},[114,4484],{"className":4485,"style":250},[191],[114,4487,1403],{"className":4488},[254],[114,4490],{"className":4491,"style":250},[191],[114,4493,4495,4499,4694,4697,4817,4860,4863,4866,4869],{"className":4494},[168],[114,4496],{"className":4497,"style":4498},[172],"height:1.44em;vertical-align:-0.5949em;",[114,4500,4502,4506,4691],{"className":4501},[181],[114,4503],{"className":4504},[177,4505],"nulldelimiter",[114,4507,4509],{"className":4508},[4369],[114,4510,4512,4682],{"className":4511},[2014,2015],[114,4513,4515,4679],{"className":4514},[2019],[114,4516,4519,4653,4664],{"className":4517,"style":4518},[2023],"height:0.8451em;",[114,4520,4522,4526],{"style":4521},"top:-2.655em;",[114,4523],{"className":4524,"style":4525},[2031],"height:3em;",[114,4527,4529],{"className":4528},[2036,2037,2038,2039],[114,4530,4532],{"className":4531},[181,2039],[114,4533,4535,4538],{"className":4534},[181,2039],[114,4536,1956],{"className":4537},[181,182,2039],[114,4539,4541],{"className":4540},[2010],[114,4542,4544,4644],{"className":4543},[2014,2015],[114,4545,4547,4641],{"className":4546},[2019],[114,4548,4551],{"className":4549,"style":4550},[2023],"height:0.3281em;",[114,4552,4553,4556],{"style":2251},[114,4554],{"className":4555,"style":2255},[2031],[114,4557,4559],{"className":4558},[2036,2259,2260,2039],[114,4560,4562,4603],{"className":4561},[181,2039],[114,4563,4565,4568],{"className":4564},[181,2039],[114,4566,1961],{"className":4567},[181,182,2039],[114,4569,4571],{"className":4570},[2010],[114,4572,4574,4594],{"className":4573},[2014,2015],[114,4575,4577,4591],{"className":4576},[2019],[114,4578,4581],{"className":4579,"style":4580},[2023],"height:0.3448em;",[114,4582,4584,4588],{"style":4583},"top:-2.3448em;margin-left:0em;margin-right:0.1em;",[114,4585],{"className":4586,"style":4587},[2031],"height:2.6444em;",[114,4589,629],{"className":4590},[181,2039],[114,4592,2052],{"className":4593},[2051],[114,4595,4597],{"className":4596},[2019],[114,4598,4601],{"className":4599,"style":4600},[2023],"height:0.2996em;",[114,4602],{},[114,4604,4606,4609],{"className":4605},[181,2039],[114,4607,1964],{"className":4608,"style":2046},[181,182,2039],[114,4610,4612],{"className":4611},[2010],[114,4613,4615,4633],{"className":4614},[2014,2015],[114,4616,4618,4630],{"className":4617},[2019],[114,4619,4621],{"className":4620,"style":4580},[2023],[114,4622,4624,4627],{"style":4623},"top:-2.3448em;margin-left:-0.0572em;margin-right:0.1em;",[114,4625],{"className":4626,"style":4587},[2031],[114,4628,629],{"className":4629},[181,2039],[114,4631,2052],{"className":4632},[2051],[114,4634,4636],{"className":4635},[2019],[114,4637,4639],{"className":4638,"style":4600},[2023],[114,4640],{},[114,4642,2052],{"className":4643},[2051],[114,4645,4647],{"className":4646},[2019],[114,4648,4651],{"className":4649,"style":4650},[2023],"height:0.357em;",[114,4652],{},[114,4654,4656,4659],{"style":4655},"top:-3.23em;",[114,4657],{"className":4658,"style":4525},[2031],[114,4660],{"className":4661,"style":4663},[4662],"frac-line","border-bottom-width:0.04em;",[114,4665,4667,4670],{"style":4666},"top:-3.394em;",[114,4668],{"className":4669,"style":4525},[2031],[114,4671,4673],{"className":4672},[2036,2037,2038,2039],[114,4674,4676],{"className":4675},[181,2039],[114,4677,4373],{"className":4678},[181,2039],[114,4680,2052],{"className":4681},[2051],[114,4683,4685],{"className":4684},[2019],[114,4686,4689],{"className":4687,"style":4688},[2023],"height:0.5949em;",[114,4690],{},[114,4692],{"className":4693},[208,4505],[114,4695,136],{"className":4696},[177],[114,4698,4700,4703],{"className":4699},[181],[114,4701,1956],{"className":4702},[181,182],[114,4704,4706],{"className":4705},[2010],[114,4707,4709,4809],{"className":4708},[2014,2015],[114,4710,4712,4806],{"className":4711},[2019],[114,4713,4715],{"className":4714,"style":2024},[2023],[114,4716,4717,4720],{"style":2027},[114,4718],{"className":4719,"style":2032},[2031],[114,4721,4723],{"className":4722},[2036,2037,2038,2039],[114,4724,4726,4766],{"className":4725},[181,2039],[114,4727,4729,4732],{"className":4728},[181,2039],[1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",[114,4885,4887,4909],{"className":4886},[117],[114,4888,4890],{"className":4889},[121],[123,4891,4892],{"xmlns":125},[127,4893,4894,4906],{},[130,4895,4896],{},[1952,4897,4898,4900],{},[138,4899,2398],{},[130,4901,4902,4904],{},[138,4903,1961],{},[138,4905,1964],{},[157,4907,4908],{"encoding":159},"E_{ij}",[114,4910,4912],{"className":4911,"ariaHidden":143},[164],[114,4913,4915,4918],{"className":4914},[168],[114,4916],{"className":4917,"style":2418},[172],[114,4919,4921,4924],{"className":4920},[181],[114,4922,2398],{"className":4923,"style":246},[181,182],[114,4925,4927],{"className":4926},[2010],[114,4928,4930,4953],{"className":4929},[2014,2015],[114,4931,4933,4950],{"className":4932},[2019],[114,4934,4936],{"className":4935,"style":2024},[2023],[114,4937,4938,4941],{"style":2440},[114,4939],{"className":4940,"style":2032},[2031],[114,4942,4944],{"className":4943},[2036,2037,2038,2039],[114,4945,4947],{"className":4946},[181,2039],[114,4948,2047],{"className":4949,"style":2046},[181,182,2039],[114,4951,2052],{"className":4952},[2051],[114,4954,4956],{"className":4955},[2019],[114,4957,4959],{"className":4958,"style":2059},[2023],[114,4960],{}," 都在 ",[114,4963,4965,4978],{"className":4964},[117],[114,4966,4968],{"className":4967},[121],[123,4969,4970],{"xmlns":125},[127,4971,4972,4976],{},[130,4973,4974],{},[138,4975,756],{},[157,4977,756],{"encoding":159},[114,4979,4981],{"className":4980,"ariaHidden":143},[164],[114,4982,4984,4987],{"className":4983},[168],[114,4985],{"className":4986,"style":264},[172],[114,4988,756],{"className":4989,"style":771},[181,182]," 中。由于任意矩阵 ",[114,4992,4994,5007],{"className":4993},[117],[114,4995,4997],{"className":4996},[121],[123,4998,4999],{"xmlns":125},[127,5000,5001,5005],{},[130,5002,5003],{},[138,5004,1476],{},[157,5006,1476],{"encoding":159},[114,5008,5010],{"className":5009,"ariaHidden":143},[164],[114,5011,5013,5016],{"className":5012},[168],[114,5014],{"className":5015,"style":264},[172],[114,5017,1476],{"className":5018,"style":1503},[181,182]," 都是 ",[114,5021,5023,5044],{"className":5022},[117],[114,5024,5026],{"className":5025},[121],[123,5027,5028],{"xmlns":125},[127,5029,5030,5042],{},[130,5031,5032],{},[1952,5033,5034,5036],{},[138,5035,2398],{},[130,5037,5038,5040],{},[138,5039,1961],{},[138,5041,1964],{},[157,5043,4908],{"encoding":159},[114,5045,5047],{"className":5046,"ariaHidden":143},[164],[114,5048,5050,5053],{"className":5049},[168],[114,5051],{"className":5052,"style":2418},[172],[114,5054,5056,5059],{"className":5055},[181],[114,5057,2398],{"className":5058,"style":246},[181,182],[114,5060,5062],{"className":5061},[2010],[114,5063,5065,5088],{"className":5064},[2014,2015],[114,5066,5068,5085],{"className":5067},[2019],[114,5069,5071],{"className":5070,"style":2024},[2023],[114,5072,5073,5076],{"style":2440},[114,5074],{"className":5075,"style":2032},[2031],[114,5077,5079],{"className":5078},[2036,2037,2038,2039],[114,5080,5082],{"className":5081},[181,2039],[114,5083,2047],{"className":5084,"style":2046},[181,182,2039],[114,5086,2052],{"className":5087},[2051],[114,5089,5091],{"className":5090},[2019],[114,5092,5094],{"className":5093,"style":2059},[2023],[114,5095],{}," 的线性组合，所以整个环 ",[114,5098,5100,5128],{"className":5099},[117],[114,5101,5103],{"className":5102},[121],[123,5104,5105],{"xmlns":125},[127,5106,5107,5125],{},[130,5108,5109,5111,5113,5115,5117,5119,5121,5123],{},[138,5110,1476],{},[133,5112,136],{"stretchy":135},[138,5114,1481],{},[133,5116,144],{"separator":143},[138,5118,140],{"mathvariant":555},[133,5120,155],{"stretchy":135},[133,5122,227],{},[138,5124,756],{},[157,5126,5127],{"encoding":159},"M(n, \\mathbb{R}) \\subseteq I",[114,5129,5131,5167],{"className":5130,"ariaHidden":143},[164],[114,5132,5134,5137,5140,5143,5146,5149,5152,5155,5158,5161,5164],{"className":5133},[168],[114,5135],{"className":5136,"style":173},[172],[114,5138,1476],{"className":5139,"style":1503},[181,182],[114,5141,136],{"className":5142},[177],[114,5144,1481],{"className":5145},[181,182],[114,5147,144],{"className":5148},[187],[114,5150],{"className":5151,"style":192},[191],[114,5153,140],{"className":5154},[181,576],[114,5156,155],{"className":5157},[208],[114,5159],{"className":5160,"style":250},[191],[114,5162,227],{"className":5163},[254],[114,5165],{"className":5166,"style":250},[191],[114,5168,5170,5173],{"className":5169},[168],[114,5171],{"className":5172,"style":264},[172],[114,5174,756],{"className":5175,"style":771},[181,182],[450,5177,5178,5179,5257],{},"故 ",[114,5180,5182,5209],{"className":5181},[117],[114,5183,5185],{"className":5184},[121],[123,5186,5187],{"xmlns":125},[127,5188,5189,5207],{},[130,5190,5191,5193,5195,5197,5199,5201,5203,5205],{},[138,5192,756],{},[133,5194,1403],{},[138,5196,1476],{},[133,5198,136],{"stretchy":135},[138,5200,1481],{},[133,5202,144],{"separator":143},[138,5204,140],{"mathvariant":555},[133,5206,155],{"stretchy":135},[157,5208,1720],{"encoding":159},[114,5210,5212,5230],{"className":5211,"ariaHidden":143},[164],[114,5213,5215,5218,5221,5224,5227],{"className":5214},[168],[114,5216],{"className":5217,"style":264},[172],[114,5219,756],{"className":5220,"style":771},[181,182],[114,5222],{"className":5223,"style":250},[191],[114,5225,1403],{"className":5226},[254],[114,5228],{"className":5229,"style":250},[191],[114,5231,5233,5236,5239,5242,5245,5248,5251,5254],{"className":5232},[168],[114,5234],{"className":5235,"style":173},[172],[114,5237,1476],{"className":5238,"style":1503},[181,182],[114,5240,136],{"className":5241},[177],[114,5243,1481],{"className":5244},[181,182],[114,5246,144],{"className":5247},[187],[114,5249],{"className":5250,"style":192},[191],[114,5252,140],{"className":5253},[181,576],[114,5255,155],{"className":5256},[208],"。证毕。",[19,5259,5260],{},[51,5261,5262],{},"作业补充（非平凡的左\u002F右理想）：",[447,5264,5265],{},[450,5266,5267,5270,5271,5306,5307,1595,5336],{},[51,5268,5269],{},"左理想例子","：所有",[51,5272,5273,5274,5305],{},"第 ",[114,5275,5277,5291],{"className":5276},[117],[114,5278,5280],{"className":5279},[121],[123,5281,5282],{"xmlns":125},[127,5283,5284,5289],{},[130,5285,5286],{},[138,5287,5288],{},"k",[157,5290,5288],{"encoding":159},[114,5292,5294],{"className":5293,"ariaHidden":143},[164],[114,5295,5297,5301],{"className":5296},[168],[114,5298],{"className":5299,"style":5300},[172],"height:0.6944em;",[114,5302,5288],{"className":5303,"style":5304},[181,182],"margin-right:0.0315em;"," 列全为 0"," 的矩阵集合 ",[114,5308,5310,5324],{"className":5309},[117],[114,5311,5313],{"className":5312},[121],[123,5314,5315],{"xmlns":125},[127,5316,5317,5322],{},[130,5318,5319],{},[138,5320,5321],{},"L",[157,5323,5321],{"encoding":159},[114,5325,5327],{"className":5326,"ariaHidden":143},[164],[114,5328,5330,5333],{"className":5329},[168],[114,5331],{"className":5332,"style":264},[172],[114,5334,5321],{"className":5335},[181,182],[447,5337,5338,5484,5868,6101,6223],{},[450,5339,5340,5343,5344,5397,5398,2779,5426,5454,5455,5483],{},[533,5341,5342],{},"验证","：设 ",[114,5345,5347,5366],{"className":5346},[117],[114,5348,5350],{"className":5349},[121],[123,5351,5352],{"xmlns":125},[127,5353,5354,5363],{},[130,5355,5356,5359,5361],{},[138,5357,5358],{},"B",[133,5360,828],{},[138,5362,5321],{},[157,5364,5365],{"encoding":159},"B \\in L",[114,5367,5369,5388],{"className":5368,"ariaHidden":143},[164],[114,5370,5372,5375,5379,5382,5385],{"className":5371},[168],[114,5373],{"className":5374,"style":1910},[172],[114,5376,5358],{"className":5377,"style":5378},[181,182],"margin-right:0.0502em;",[114,5380],{"className":5381,"style":250},[191],[114,5383,828],{"className":5384},[254],[114,5386],{"className":5387,"style":250},[191],[114,5389,5391,5394],{"className":5390},[168],[114,5392],{"className":5393,"style":264},[172],[114,5395,5321],{"className":5396},[181,182],"，即 ",[114,5399,5401,5414],{"className":5400},[117],[114,5402,5404],{"className":5403},[121],[123,5405,5406],{"xmlns":125},[127,5407,5408,5412],{},[130,5409,5410],{},[138,5411,5358],{},[157,5413,5358],{"encoding":159},[114,5415,5417],{"className":5416,"ariaHidden":143},[164],[114,5418,5420,5423],{"className":5419},[168],[114,5421],{"className":5422,"style":264},[172],[114,5424,5358],{"className":5425,"style":5378},[181,182],[114,5427,5429,5442],{"className":5428},[117],[114,5430,5432],{"className":5431},[121],[123,5433,5434],{"xmlns":125},[127,5435,5436,5440],{},[130,5437,5438],{},[138,5439,5288],{},[157,5441,5288],{"encoding":159},[114,5443,5445],{"className":5444,"ariaHidden":143},[164],[114,5446,5448,5451],{"className":5447},[168],[114,5449],{"className":5450,"style":5300},[172],[114,5452,5288],{"className":5453,"style":5304},[181,182]," 列为 0。设 ",[114,5456,5458,5471],{"className":5457},[117],[114,5459,5461],{"className":5460},[121],[123,5462,5463],{"xmlns":125},[127,5464,5465,5469],{},[130,5466,5467],{},[138,5468,1893],{},[157,5470,1893],{"encoding":159},[114,5472,5474],{"className":5473,"ariaHidden":143},[164],[114,5475,5477,5480],{"className":5476},[168],[114,5478],{"className":5479,"style":264},[172],[114,5481,1893],{"className":5482},[181,182]," 是任意矩阵。",[450,5485,5486,5487,445,5545,2779,5573,5601,5602,445],{},"考虑乘积 ",[114,5488,5490,5511],{"className":5489},[117],[114,5491,5493],{"className":5492},[121],[123,5494,5495],{"xmlns":125},[127,5496,5497,5508],{},[130,5498,5499,5502,5504,5506],{},[138,5500,5501],{},"C",[133,5503,1403],{},[138,5505,1893],{},[138,5507,5358],{},[157,5509,5510],{"encoding":159},"C = AB",[114,5512,5514,5533],{"className":5513,"ariaHidden":143},[164],[114,5515,5517,5520,5524,5527,5530],{"className":5516},[168],[114,5518],{"className":5519,"style":264},[172],[114,5521,5501],{"className":5522,"style":5523},[181,182],"margin-right:0.0715em;",[114,5525],{"className":5526,"style":250},[191],[114,5528,1403],{"className":5529},[254],[114,5531],{"className":5532,"style":250},[191],[114,5534,5536,5539,5542],{"className":5535},[168],[114,5537],{"className":5538,"style":264},[172],[114,5540,1893],{"className":5541},[181,182],[114,5543,5358],{"className":5544,"style":5378},[181,182],[114,5546,5548,5561],{"className":5547},[117],[114,5549,5551],{"className":5550},[121],[123,5552,5553],{"xmlns":125},[127,5554,5555,5559],{},[130,5556,5557],{},[138,5558,5501],{},[157,5560,5501],{"encoding":159},[114,5562,5564],{"className":5563,"ariaHidden":143},[164],[114,5565,5567,5570],{"className":5566},[168],[114,5568],{"className":5569,"style":264},[172],[114,5571,5501],{"className":5572,"style":5523},[181,182],[114,5574,5576,5589],{"className":5575},[117],[114,5577,5579],{"className":5578},[121],[123,5580,5581],{"xmlns":125},[127,5582,5583,5587],{},[130,5584,5585],{},[138,5586,5288],{},[157,5588,5288],{"encoding":159},[114,5590,5592],{"className":5591,"ariaHidden":143},[164],[114,5593,5595,5598],{"className":5594},[168],[114,5596],{"className":5597,"style":5300},[172],[114,5599,5288],{"className":5600,"style":5304},[181,182]," 列元素为 ",[114,5603,5605,5658],{"className":5604},[117],[114,5606,5608],{"className":5607},[121],[123,5609,5610],{"xmlns":125},[127,5611,5612,5655],{},[130,5613,5614,5625,5627,5634,5644],{},[1952,5615,5616,5619],{},[138,5617,5618],{},"c",[130,5620,5621,5623],{},[138,5622,1961],{},[138,5624,5288],{},[133,5626,1403],{},[1952,5628,5629,5632],{},[133,5630,5631],{},"∑",[138,5633,1964],{},[1952,5635,5636,5638],{},[138,5637,1956],{},[130,5639,5640,5642],{},[138,5641,1961],{},[138,5643,1964],{},[1952,5645,5646,5649],{},[138,5647,5648],{},"b",[130,5650,5651,5653],{},[138,5652,1964],{},[138,5654,5288],{},[157,5656,5657],{"encoding":159},"c_{ik} = \\sum_j a_{ij} b_{jk}",[114,5659,5661,5722],{"className":5660,"ariaHidden":143},[164],[114,5662,5664,5668,5713,5716,5719],{"className":5663},[168],[114,5665],{"className":5666,"style":5667},[172],"height:0.5806em;vertical-align:-0.15em;",[114,5669,5671,5674],{"className":5670},[181],[114,5672,5618],{"className":5673},[181,182],[114,5675,5677],{"className":5676},[2010],[114,5678,5680,5705],{"className":5679},[2014,2015],[114,5681,5683,5702],{"className":5682},[2019],[114,5684,5687],{"className":5685,"style":5686},[2023],"height:0.3361em;",[114,5688,5689,5692],{"style":2027},[114,5690],{"className":5691,"style":2032},[2031],[114,5693,5695],{"className":5694},[2036,2037,2038,2039],[114,5696,5698],{"className":5697},[181,2039],[114,5699,5701],{"className":5700,"style":5304},[181,182,2039],"ik",[114,5703,2052],{"className":5704},[2051],[114,5706,5708],{"className":5707},[2019],[114,5709,5711],{"className":5710,"style":2850},[2023],[114,5712],{},[114,5714],{"className":5715,"style":250},[191],[114,5717,1403],{"className":5718},[254],[114,5720],{"className":5721,"style":250},[191],[114,5723,5725,5729,5776,5779,5822],{"className":5724},[168],[114,5726],{"className":5727,"style":5728},[172],"height:1.1858em;vertical-align:-0.4358em;",[114,5730,5733,5739],{"className":5731},[5732],"mop",[114,5734,5631],{"className":5735,"style":5738},[5732,5736,5737],"op-symbol","small-op","position:relative;top:0em;",[114,5740,5742],{"className":5741},[2010],[114,5743,5745,5767],{"className":5744},[2014,2015],[114,5746,5748,5764],{"className":5747},[2019],[114,5749,5752],{"className":5750,"style":5751},[2023],"height:0.162em;",[114,5753,5755,5758],{"style":5754},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[114,5756],{"className":5757,"style":2032},[2031],[114,5759,5761],{"className":5760},[2036,2037,2038,2039],[114,5762,1964],{"className":5763,"style":2046},[181,182,2039],[114,5765,2052],{"className":5766},[2051],[114,5768,5770],{"className":5769},[2019],[114,5771,5774],{"className":5772,"style":5773},[2023],"height:0.4358em;",[114,5775],{},[114,5777],{"className":5778,"style":192},[191],[114,5780,5782,5785],{"className":5781},[181],[114,5783,1956],{"className":5784},[181,182],[114,5786,5788],{"className":5787},[2010],[114,5789,5791,5814],{"className":5790},[2014,2015],[114,5792,5794,5811],{"className":5793},[2019],[114,5795,5797],{"className":5796,"style":2024},[2023],[114,5798,5799,5802],{"style":2027},[114,5800],{"className":5801,"style":2032},[2031],[114,5803,5805],{"className":5804},[2036,2037,2038,2039],[114,5806,5808],{"className":5807},[181,2039],[114,5809,2047],{"className":5810,"style":2046},[181,182,2039],[114,5812,2052],{"className":5813},[2051],[114,5815,5817],{"className":5816},[2019],[114,5818,5820],{"className":5819,"style":2059},[2023],[114,5821],{},[114,5823,5825,5828],{"className":5824},[181],[114,5826,5648],{"className":5827},[181,182],[114,5829,5831],{"className":5830},[2010],[114,5832,5834,5860],{"className":5833},[2014,2015],[114,5835,5837,5857],{"className":5836},[2019],[114,5838,5840],{"className":5839,"style":5686},[2023],[114,5841,5842,5845],{"style":2027},[114,5843],{"className":5844,"style":2032},[2031],[114,5846,5848],{"className":5847},[2036,2037,2038,2039],[114,5849,5851,5854],{"className":5850},[181,2039],[114,5852,1964],{"className":5853,"style":2046},[181,182,2039],[114,5855,5288],{"className":5856,"style":5304},[181,182,2039],[114,5858,2052],{"className":5859},[2051],[114,5861,5863],{"className":5862},[2019],[114,5864,5866],{"className":5865,"style":2059},[2023],[114,5867],{},[450,5869,1773,5870,5972,5973,6001,6002,445],{},[114,5871,5873,5899],{"className":5872},[117],[114,5874,5876],{"className":5875},[121],[123,5877,5878],{"xmlns":125},[127,5879,5880,5896],{},[130,5881,5882,5892,5894],{},[1952,5883,5884,5886],{},[138,5885,5648],{},[130,5887,5888,5890],{},[138,5889,1964],{},[138,5891,5288],{},[133,5893,1403],{},[550,5895,629],{},[157,5897,5898],{"encoding":159},"b_{jk} = 0",[114,5900,5902,5963],{"className":5901,"ariaHidden":143},[164],[114,5903,5905,5908,5954,5957,5960],{"className":5904},[168],[114,5906],{"className":5907,"style":2200},[172],[114,5909,5911,5914],{"className":5910},[181],[114,5912,5648],{"className":5913},[181,182],[114,5915,5917],{"className":5916},[2010],[114,5918,5920,5946],{"className":5919},[2014,2015],[114,5921,5923,5943],{"className":5922},[2019],[114,5924,5926],{"className":5925,"style":5686},[2023],[114,5927,5928,5931],{"style":2027},[114,5929],{"className":5930,"style":2032},[2031],[114,5932,5934],{"className":5933},[2036,2037,2038,2039],[114,5935,5937,5940],{"className":5936},[181,2039],[114,5938,1964],{"className":5939,"style":2046},[181,182,2039],[114,5941,5288],{"className":5942,"style":5304},[181,182,2039],[114,5944,2052],{"className":5945},[2051],[114,5947,5949],{"className":5948},[2019],[114,5950,5952],{"className":5951,"style":2059},[2023],[114,5953],{},[114,5955],{"className":5956,"style":250},[191],[114,5958,1403],{"className":5959},[254],[114,5961],{"className":5962,"style":250},[191],[114,5964,5966,5969],{"className":5965},[168],[114,5967],{"className":5968,"style":2149},[172],[114,5970,629],{"className":5971},[181],"（对所有 ",[114,5974,5976,5989],{"className":5975},[117],[114,5977,5979],{"className":5978},[121],[123,5980,5981],{"xmlns":125},[127,5982,5983,5987],{},[130,5984,5985],{},[138,5986,1964],{},[157,5988,1964],{"encoding":159},[114,5990,5992],{"className":5991,"ariaHidden":143},[164],[114,5993,5995,5998],{"className":5994},[168],[114,5996],{"className":5997,"style":2914},[172],[114,5999,1964],{"className":6000,"style":2046},[181,182],"），所以 ",[114,6003,6005,6031],{"className":6004},[117],[114,6006,6008],{"className":6007},[121],[123,6009,6010],{"xmlns":125},[127,6011,6012,6028],{},[130,6013,6014,6024,6026],{},[1952,6015,6016,6018],{},[138,6017,5618],{},[130,6019,6020,6022],{},[138,6021,1961],{},[138,6023,5288],{},[133,6025,1403],{},[550,6027,629],{},[157,6029,6030],{"encoding":159},"c_{ik} = 0",[114,6032,6034,6092],{"className":6033,"ariaHidden":143},[164],[114,6035,6037,6040,6083,6086,6089],{"className":6036},[168],[114,6038],{"className":6039,"style":5667},[172],[114,6041,6043,6046],{"className":6042},[181],[114,6044,5618],{"className":6045},[181,182],[114,6047,6049],{"className":6048},[2010],[114,6050,6052,6075],{"className":6051},[2014,2015],[114,6053,6055,6072],{"className":6054},[2019],[114,6056,6058],{"className":6057,"style":5686},[2023],[114,6059,6060,6063],{"style":2027},[114,6061],{"className":6062,"style":2032},[2031],[114,6064,6066],{"className":6065},[2036,2037,2038,2039],[114,6067,6069],{"className":6068},[181,2039],[114,6070,5701],{"className":6071,"style":5304},[181,182,2039],[114,6073,2052],{"className":6074},[2051],[114,6076,6078],{"className":6077},[2019],[114,6079,6081],{"className":6080,"style":2850},[2023],[114,6082],{},[114,6084],{"className":6085,"style":250},[191],[114,6087,1403],{"className":6088},[254],[114,6090],{"className":6091,"style":250},[191],[114,6093,6095,6098],{"className":6094},[168],[114,6096],{"className":6097,"style":2149},[172],[114,6099,629],{"className":6100},[181],[450,6102,6103,6104,2779,6138,6166,6167,445],{},"即 ",[114,6105,6107,6123],{"className":6106},[117],[114,6108,6110],{"className":6109},[121],[123,6111,6112],{"xmlns":125},[127,6113,6114,6120],{},[130,6115,6116,6118],{},[138,6117,1893],{},[138,6119,5358],{},[157,6121,6122],{"encoding":159},"AB",[114,6124,6126],{"className":6125,"ariaHidden":143},[164],[114,6127,6129,6132,6135],{"className":6128},[168],[114,6130],{"className":6131,"style":264},[172],[114,6133,1893],{"className":6134},[181,182],[114,6136,5358],{"className":6137,"style":5378},[181,182],[114,6139,6141,6154],{"className":6140},[117],[114,6142,6144],{"className":6143},[121],[123,6145,6146],{"xmlns":125},[127,6147,6148,6152],{},[130,6149,6150],{},[138,6151,5288],{},[157,6153,5288],{"encoding":159},[114,6155,6157],{"className":6156,"ariaHidden":143},[164],[114,6158,6160,6163],{"className":6159},[168],[114,6161],{"className":6162,"style":5300},[172],[114,6164,5288],{"className":6165,"style":5304},[181,182]," 列也是 0，所以 ",[114,6168,6170,6190],{"className":6169},[117],[114,6171,6173],{"className":6172},[121],[123,6174,6175],{"xmlns":125},[127,6176,6177,6187],{},[130,6178,6179,6181,6183,6185],{},[138,6180,1893],{},[138,6182,5358],{},[133,6184,828],{},[138,6186,5321],{},[157,6188,6189],{"encoding":159},"AB \\in L",[114,6191,6193,6214],{"className":6192,"ariaHidden":143},[164],[114,6194,6196,6199,6202,6205,6208,6211],{"className":6195},[168],[114,6197],{"className":6198,"style":1910},[172],[114,6200,1893],{"className":6201},[181,182],[114,6203,5358],{"className":6204,"style":5378},[181,182],[114,6206],{"className":6207,"style":250},[191],[114,6209,828],{"className":6210},[254],[114,6212],{"className":6213,"style":250},[191],[114,6215,6217,6220],{"className":6216},[168],[114,6218],{"className":6219,"style":264},[172],[114,6221,5321],{"className":6222},[181,182],[450,6224,6225,6226,6254,6255,6283],{},"所以 ",[114,6227,6229,6242],{"className":6228},[117],[114,6230,6232],{"className":6231},[121],[123,6233,6234],{"xmlns":125},[127,6235,6236,6240],{},[130,6237,6238],{},[138,6239,5321],{},[157,6241,5321],{"encoding":159},[114,6243,6245],{"className":6244,"ariaHidden":143},[164],[114,6246,6248,6251],{"className":6247},[168],[114,6249],{"className":6250,"style":264},[172],[114,6252,5321],{"className":6253},[181,182]," 是左理想。但它不是右理想（右乘一个矩阵可以将其他列的数据加到第 ",[114,6256,6258,6271],{"className":6257},[117],[114,6259,6261],{"className":6260},[121],[123,6262,6263],{"xmlns":125},[127,6264,6265,6269],{},[130,6266,6267],{},[138,6268,5288],{},[157,6270,5288],{"encoding":159},[114,6272,6274],{"className":6273,"ariaHidden":143},[164],[114,6275,6277,6280],{"className":6276},[168],[114,6278],{"className":6279,"style":5300},[172],[114,6281,5288],{"className":6282,"style":5304},[181,182]," 列）。",[99,6285],{},[14,6287,6289],{"id":6288},"_4-由子集生成的理想-generated-ideals","4. 由子集生成的理想 (Generated Ideals)",[106,6291,6293],{"id":6292},"_41-定义与交集性质","4.1 定义与交集性质",[19,6295,6296,5343,6299,6349,6350,6401,6402,6434,6435,6463,6464,6517,6518,6569,6570,6620,6621,6624,6625,445],{},[51,6297,6298],{},"定义 3.2.4",[114,6300,6302,6319],{"className":6301},[117],[114,6303,6305],{"className":6304},[121],[123,6306,6307],{"xmlns":125},[127,6308,6309,6317],{},[130,6310,6311,6313,6315],{},[138,6312,224],{},[133,6314,227],{},[138,6316,140],{},[157,6318,232],{"encoding":159},[114,6320,6322,6340],{"className":6321,"ariaHidden":143},[164],[114,6323,6325,6328,6331,6334,6337],{"className":6324},[168],[114,6326],{"className":6327,"style":242},[172],[114,6329,224],{"className":6330,"style":246},[181,182],[114,6332],{"className":6333,"style":250},[191],[114,6335,227],{"className":6336},[254],[114,6338],{"className":6339,"style":250},[191],[114,6341,6343,6346],{"className":6342},[168],[114,6344],{"className":6345,"style":264},[172],[114,6347,140],{"className":6348,"style":183},[181,182],"（或记为 ",[114,6351,6353,6371],{"className":6352},[117],[114,6354,6356],{"className":6355},[121],[123,6357,6358],{"xmlns":125},[127,6359,6360,6368],{},[130,6361,6362,6364,6366],{},[138,6363,1476],{},[133,6365,227],{},[138,6367,140],{},[157,6369,6370],{"encoding":159},"M \\subseteq R",[114,6372,6374,6392],{"className":6373,"ariaHidden":143},[164],[114,6375,6377,6380,6383,6386,6389],{"className":6376},[168],[114,6378],{"className":6379,"style":242},[172],[114,6381,1476],{"className":6382,"style":1503},[181,182],[114,6384],{"className":6385,"style":250},[191],[114,6387,227],{"className":6388},[254],[114,6390],{"className":6391,"style":250},[191],[114,6393,6395,6398],{"className":6394},[168],[114,6396],{"className":6397,"style":264},[172],[114,6399,140],{"className":6400,"style":183},[181,182],"）。",[51,6403,6404,6405,6433],{},"包含 ",[114,6406,6408,6421],{"className":6407},[117],[114,6409,6411],{"className":6410},[121],[123,6412,6413],{"xmlns":125},[127,6414,6415,6419],{},[130,6416,6417],{},[138,6418,224],{},[157,6420,224],{"encoding":159},[114,6422,6424],{"className":6423,"ariaHidden":143},[164],[114,6425,6427,6430],{"className":6426},[168],[114,6428],{"className":6429,"style":264},[172],[114,6431,224],{"className":6432,"style":246},[181,182]," 的最小的理想","称为由 ",[114,6436,6438,6451],{"className":6437},[117],[114,6439,6441],{"className":6440},[121],[123,6442,6443],{"xmlns":125},[127,6444,6445,6449],{},[130,6446,6447],{},[138,6448,224],{},[157,6450,224],{"encoding":159},[114,6452,6454],{"className":6453,"ariaHidden":143},[164],[114,6455,6457,6460],{"className":6456},[168],[114,6458],{"className":6459,"style":264},[172],[114,6461,224],{"className":6462,"style":246},[181,182]," 生成的理想，记为 ",[114,6465,6467,6487],{"className":6466},[117],[114,6468,6470],{"className":6469},[121],[123,6471,6472],{"xmlns":125},[127,6473,6474,6484],{},[130,6475,6476,6479,6481],{},[133,6477,6478],{},"\u003C",[138,6480,224],{},[133,6482,6483],{},">",[157,6485,6486],{"encoding":159},"\u003CS>",[114,6488,6490,6502],{"className":6489,"ariaHidden":143},[164],[114,6491,6493,6496,6499],{"className":6492},[168],[114,6494],{"className":6495,"style":852},[172],[114,6497,6478],{"className":6498},[254],[114,6500],{"className":6501,"style":250},[191],[114,6503,6505,6508,6511,6514],{"className":6504},[168],[114,6506],{"className":6507,"style":1910},[172],[114,6509,224],{"className":6510,"style":246},[181,182],[114,6512],{"className":6513,"style":250},[191],[114,6515,6483],{"className":6516},[254]," 或 ",[114,6519,6521,6539],{"className":6520},[117],[114,6522,6524],{"className":6523},[121],[123,6525,6526],{"xmlns":125},[127,6527,6528,6536],{},[130,6529,6530,6532,6534],{},[133,6531,6478],{},[138,6533,1476],{},[133,6535,6483],{},[157,6537,6538],{"encoding":159},"\u003CM>",[114,6540,6542,6554],{"className":6541,"ariaHidden":143},[164],[114,6543,6545,6548,6551],{"className":6544},[168],[114,6546],{"className":6547,"style":852},[172],[114,6549,6478],{"className":6550},[254],[114,6552],{"className":6553,"style":250},[191],[114,6555,6557,6560,6563,6566],{"className":6556},[168],[114,6558],{"className":6559,"style":1910},[172],[114,6561,1476],{"className":6562,"style":1503},[181,182],[114,6564],{"className":6565,"style":250},[191],[114,6567,6483],{"className":6568},[254],"。\n如果 ",[114,6571,6573,6590],{"className":6572},[117],[114,6574,6576],{"className":6575},[121],[123,6577,6578],{"xmlns":125},[127,6579,6580,6588],{},[130,6581,6582,6584,6586],{},[133,6583,6478],{},[138,6585,1476],{},[133,6587,6483],{},[157,6589,6538],{"encoding":159},[114,6591,6593,6605],{"className":6592,"ariaHidden":143},[164],[114,6594,6596,6599,6602],{"className":6595},[168],[114,6597],{"className":6598,"style":852},[172],[114,6600,6478],{"className":6601},[254],[114,6603],{"className":6604,"style":250},[191],[114,6606,6608,6611,6614,6617],{"className":6607},[168],[114,6609],{"className":6610,"style":1910},[172],[114,6612,1476],{"className":6613,"style":1503},[181,182],[114,6615],{"className":6616,"style":250},[191],[114,6618,6483],{"className":6619},[254]," 可以由有限集生成，称该理想是",[51,6622,6623],{},"有限生成","的。如果由一个元素生成，称为",[51,6626,6627],{},"主理想",[19,6629,6630,6633,6634,6782,6783,6811,6812,6840,6841,445],{},[51,6631,6632],{},"命题 3.2.6","（构造性定义）：\n",[114,6635,6637,6677],{"className":6636},[117],[114,6638,6640],{"className":6639},[121],[123,6641,6642],{"xmlns":125},[127,6643,6644,6674],{},[130,6645,6646,6648,6650,6652,6654,6672],{},[133,6647,6478],{},[138,6649,1476],{},[133,6651,6483],{},[133,6653,1403],{},[1952,6655,6656,6659],{},[133,6657,6658],{},"⋂",[130,6660,6661,6663,6665,6668,6670],{},[138,6662,1476],{},[133,6664,227],{},[138,6666,6667],{},"H",[133,6669,1311],{},[138,6671,140],{},[138,6673,6667],{},[157,6675,6676],{"encoding":159},"\u003C M > = \\bigcap_{M \\subseteq H \\triangleleft R} H",[114,6678,6680,6692,6711],{"className":6679,"ariaHidden":143},[164],[114,6681,6683,6686,6689],{"className":6682},[168],[114,6684],{"className":6685,"style":852},[172],[114,6687,6478],{"className":6688},[254],[114,6690],{"className":6691,"style":250},[191],[114,6693,6695,6698,6701,6704,6708],{"className":6694},[168],[114,6696],{"className":6697,"style":1910},[172],[114,6699,1476],{"className":6700,"style":1503},[181,182],[114,6702],{"className":6703,"style":250},[191],[114,6705,6707],{"className":6706},[254],">=",[114,6709],{"className":6710,"style":250},[191],[114,6712,6714,6718,6776,6779],{"className":6713},[168],[114,6715],{"className":6716,"style":6717},[172],"height:1.1449em;vertical-align:-0.3949em;",[114,6719,6721,6724],{"className":6720},[5732],[114,6722,6658],{"className":6723,"style":5738},[5732,5736,5737],[114,6725,6727],{"className":6726},[2010],[114,6728,6730,6767],{"className":6729},[2014,2015],[114,6731,6733,6764],{"className":6732},[2019],[114,6734,6737],{"className":6735,"style":6736},[2023],"height:0.1786em;",[114,6738,6739,6742],{"style":5754},[114,6740],{"className":6741,"style":2032},[2031],[114,6743,6745],{"className":6744},[2036,2037,2038,2039],[114,6746,6748,6751,6754,6758,6761],{"className":6747},[181,2039],[114,6749,1476],{"className":6750,"style":1503},[181,182,2039],[114,6752,227],{"className":6753},[254,2039],[114,6755,6667],{"className":6756,"style":6757},[181,182,2039],"margin-right:0.0813em;",[114,6759,1311],{"className":6760},[656,2039],[114,6762,140],{"className":6763,"style":183},[181,182,2039],[114,6765,2052],{"className":6766},[2051],[114,6768,6770],{"className":6769},[2019],[114,6771,6774],{"className":6772,"style":6773},[2023],"height:0.3949em;",[114,6775],{},[114,6777],{"className":6778,"style":192},[191],[114,6780,6667],{"className":6781,"style":6757},[181,182],"\n即：所有包含 ",[114,6784,6786,6799],{"className":6785},[117],[114,6787,6789],{"className":6788},[121],[123,6790,6791],{"xmlns":125},[127,6792,6793,6797],{},[130,6794,6795],{},[138,6796,1476],{},[157,6798,1476],{"encoding":159},[114,6800,6802],{"className":6801,"ariaHidden":143},[164],[114,6803,6805,6808],{"className":6804},[168],[114,6806],{"className":6807,"style":264},[172],[114,6809,1476],{"className":6810,"style":1503},[181,182]," 的理想的交集。特别地，如果 ",[114,6813,6815,6828],{"className":6814},[117],[114,6816,6818],{"className":6817},[121],[123,6819,6820],{"xmlns":125},[127,6821,6822,6826],{},[130,6823,6824],{},[138,6825,1476],{},[157,6827,1476],{"encoding":159},[114,6829,6831],{"className":6830,"ariaHidden":143},[164],[114,6832,6834,6837],{"className":6833},[168],[114,6835],{"className":6836,"style":264},[172],[114,6838,1476],{"className":6839,"style":1503},[181,182]," 本身就是理想，则 ",[114,6842,6844,6866],{"className":6843},[117],[114,6845,6847],{"className":6846},[121],[123,6848,6849],{"xmlns":125},[127,6850,6851,6863],{},[130,6852,6853,6855,6857,6859,6861],{},[133,6854,6478],{},[138,6856,1476],{},[133,6858,6483],{},[133,6860,1403],{},[138,6862,1476],{},[157,6864,6865],{"encoding":159},"\u003CM> = M",[114,6867,6869,6881,6899],{"className":6868,"ariaHidden":143},[164],[114,6870,6872,6875,6878],{"className":6871},[168],[114,6873],{"className":6874,"style":852},[172],[114,6876,6478],{"className":6877},[254],[114,6879],{"className":6880,"style":250},[191],[114,6882,6884,6887,6890,6893,6896],{"className":6883},[168],[114,6885],{"className":6886,"style":1910},[172],[114,6888,1476],{"className":6889,"style":1503},[181,182],[114,6891],{"className":6892,"style":250},[191],[114,6894,6707],{"className":6895},[254],[114,6897],{"className":6898,"style":250},[191],[114,6900,6902,6905],{"className":6901},[168],[114,6903],{"className":6904,"style":264},[172],[114,6906,1476],{"className":6907,"style":1503},[181,182],[106,6909,6911],{"id":6910},"_42-生成理想的元素形式公式解析","4.2 生成理想的元素形式（公式解析）",[19,6913,6914],{},"根据环的不同性质（是否有单位元、是否交换），生成理想中元素的具体样子是不同的。这是因为我们需要通过“补丁”来强行满足理想的定义（加法封闭、左右乘封闭）。",[19,6916,6917],{},[51,6918,6919],{},"情况 1：最一般的环（非交换、无单位元）",[114,6921,6924],{"className":6922},[6923],"katex-display",[114,6925,6927,7138],{"className":6926},[117],[114,6928,6930],{"className":6929},[121],[123,6931,6933],{"xmlns":125,"display":6932},"block",[127,6934,6935,7135],{},[130,6936,6937,6939,6941,6943,6945],{},[133,6938,6478],{},[138,6940,1476],{},[133,6942,6483],{},[133,6944,1403],{},[130,6946,6947,6949,6958,6960,6966,6973,6975,6981,6992,6994,7006,7014,7016,7029,7043,7057,7059,7062,7066,7068,7133],{},[133,6948,626],{"fence":143},[6950,6951,6952,6954],"munder",{},[133,6953,5631],{},[6955,6956,6957],"mtext",{},"finite",[133,6959,136],{"stretchy":135},[1952,6961,6962,6964],{},[138,6963,1956],{},[138,6965,1961],{},[1952,6967,6968,6971],{},[138,6969,6970],{},"m",[138,6972,1961],{},[133,6974,147],{},[1952,6976,6977,6979],{},[138,6978,825],{},[138,6980,1964],{},[6982,6983,6984,6986,6988],"msubsup",{},[138,6985,6970],{},[138,6987,1964],{},[133,6989,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},"0em","′",[133,6993,147],{},[6982,6995,6996,6998,7000],{},[138,6997,6970],{},[138,6999,5288],{},[130,7001,7002,7004],{},[133,7003,6991],{"mathvariant":1790},[133,7005,6991],{"mathvariant":1790},[6982,7007,7008,7010,7012],{},[138,7009,825],{},[138,7011,5288],{},[133,7013,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[133,7015,147],{},[6982,7017,7018,7020,7023],{},[138,7019,825],{},[138,7021,7022],{},"t",[130,7024,7025,7027],{},[133,7026,6991],{"mathvariant":1790},[133,7028,6991],{"mathvariant":1790},[6982,7030,7031,7033,7035],{},[138,7032,6970],{},[138,7034,7022],{},[130,7036,7037,7039,7041],{},[133,7038,6991],{"mathvariant":1790},[133,7040,6991],{"mathvariant":1790},[133,7042,6991],{"mathvariant":1790},[6982,7044,7045,7047,7049],{},[138,7046,825],{},[138,7048,7022],{},[130,7050,7051,7053,7055],{},[133,7052,6991],{"mathvariant":1790},[133,7054,6991],{"mathvariant":1790},[133,7056,6991],{"mathvariant":1790},[133,7058,155],{"stretchy":135},[6955,7060,7061],{},"  ",[133,7063,7065],{"fence":143,"lspace":7064,"rspace":7064},"0.05em","|",[6955,7067,7061],{},[7069,7070,7073,7103],"mtable",{"rowspacing":7071,"columnalign":7072,"columnspacing":6990},"0.25em","right left",[7074,7075,7076,7084],"mtr",{},[7077,7078,7079],"mtd",{},[7080,7081,7082],"mstyle",{"scriptlevel":629,"displaystyle":143},[130,7083],{},[7077,7085,7086],{},[7080,7087,7088],{"scriptlevel":629,"displaystyle":143},[130,7089,7090,7092,7094,7096,7098,7101],{},[130,7091],{},[138,7093,1956],{},[133,7095,828],{},[138,7097,556],{"mathvariant":555},[6955,7099,7100],{}," (整数倍)",[133,7102,144],{"separator":143},[7074,7104,7105,7111],{},[7077,7106,7107],{},[7080,7108,7109],{"scriptlevel":629,"displaystyle":143},[130,7110],{},[7077,7112,7113],{},[7080,7114,7115],{"scriptlevel":629,"displaystyle":143},[130,7116,7117,7119,7121,7123,7125,7127,7129,7131],{},[130,7118],{},[138,7120,6970],{},[133,7122,828],{},[138,7124,1476],{},[133,7126,144],{"separator":143},[138,7128,825],{},[133,7130,828],{},[138,7132,140],{},[133,7134,632],{"fence":143},[157,7136,7137],{"encoding":159},"\u003C M > = \\left\\{ \\sum_{\\text{finite}} (a_i m_i + r_j m_j' + m_k'' r_k' + r_t'' m_t''' r_t''') \\;\\middle|\\; \\begin{aligned} &a \\in \\mathbb{Z} \\text{ (整数倍)}, \\\\ &m \\in M, r \\in R \\end{aligned} \\right\\}",[114,7139,7141,7153,7171],{"className":7140,"ariaHidden":143},[164],[114,7142,7144,7147,7150],{"className":7143},[168],[114,7145],{"className":7146,"style":852},[172],[114,7148,6478],{"className":7149},[254],[114,7151],{"className":7152,"style":250},[191],[114,7154,7156,7159,7162,7165,7168],{"className":7155},[168],[114,7157],{"className":7158,"style":1910},[172],[114,7160,1476],{"className":7161,"style":1503},[181,182],[114,7163],{"className":7164,"style":250},[191],[114,7166,6707],{"className":7167},[254],[114,7169],{"className":7170,"style":250},[191],[114,7172,7174,7178],{"className":7173},[168],[114,7175],{"className":7176,"style":7177},[172],"height:3.0521em;vertical-align:-1.3021em;",[114,7179,7182,7192,7250,7253,7293,7333,7336,7339,7342,7383,7441,7444,7447,7450,7506,7561,7564,7567,7570,7624,7679,7733,7736,7739,7788,7791,7959],{"className":7180},[7181],"minner",[114,7183,7187],{"className":7184,"style":7186},[177,7185],"delimcenter","top:0em;",[114,7188,626],{"className":7189},[7190,7191],"delimsizing","size4",[114,7193,7196],{"className":7194},[5732,7195],"op-limits",[114,7197,7199,7241],{"className":7198},[2014,2015],[114,7200,7202,7238],{"className":7201},[2019],[114,7203,7206,7226],{"className":7204,"style":7205},[2023],"height:1.05em;",[114,7207,7209,7213],{"style":7208},"top:-1.8479em;margin-left:0em;",[114,7210],{"className":7211,"style":7212},[2031],"height:3.05em;",[114,7214,7216],{"className":7215},[2036,2037,2038,2039],[114,7217,7219],{"className":7218},[181,2039],[114,7220,7223],{"className":7221},[181,7222,2039],"text",[114,7224,6957],{"className":7225},[181,2039],[114,7227,7229,7232],{"style":7228},"top:-3.05em;",[114,7230],{"className":7231,"style":7212},[2031],[114,7233,7234],{},[114,7235,5631],{"className":7236},[5732,5736,7237],"large-op",[114,7239,2052],{"className":7240},[2051],[114,7242,7244],{"className":7243},[2019],[114,7245,7248],{"className":7246,"style":7247},[2023],"height:1.3021em;",[114,7249],{},[114,7251,136],{"className":7252},[177],[114,7254,7256,7259],{"className":7255},[181],[114,7257,1956],{"className":7258},[181,182],[114,7260,7262],{"className":7261},[2010],[114,7263,7265,7285],{"className":7264},[2014,2015],[114,7266,7268,7282],{"className":7267},[2019],[114,7269,7271],{"className":7270,"style":2024},[2023],[114,7272,7273,7276],{"style":2027},[114,7274],{"className":7275,"style":2032},[2031],[114,7277,7279],{"className":7278},[2036,2037,2038,2039],[114,7280,1961],{"className":7281},[181,182,2039],[114,7283,2052],{"className":7284},[2051],[114,7286,7288],{"className":7287},[2019],[114,7289,7291],{"className":7290,"style":2850},[2023],[114,7292],{},[114,7294,7296,7299],{"className":7295},[181],[114,7297,6970],{"className":7298},[181,182],[114,7300,7302],{"className":7301},[2010],[114,7303,7305,7325],{"className":7304},[2014,2015],[114,7306,7308,7322],{"className":7307},[2019],[114,7309,7311],{"className":7310,"style":2024},[2023],[114,7312,7313,7316],{"style":2027},[114,7314],{"className":7315,"style":2032},[2031],[114,7317,7319],{"className":7318},[2036,2037,2038,2039],[114,7320,1961],{"className":7321},[181,182,2039],[114,7323,2052],{"className":7324},[2051],[114,7326,7328],{"className":7327},[2019],[114,7329,7331],{"className":7330,"style":2850},[2023],[114,7332],{},[114,7334],{"className":7335,"style":652},[191],[114,7337,147],{"className":7338},[656],[114,7340],{"className":7341,"style":652},[191],[114,7343,7345,7348],{"className":7344},[181],[114,7346,825],{"className":7347,"style":856},[181,182],[114,7349,7351],{"className":7350},[2010],[114,7352,7354,7375],{"className":7353},[2014,2015],[114,7355,7357,7372],{"className":7356},[2019],[114,7358,7360],{"className":7359,"style":2024},[2023],[114,7361,7363,7366],{"style":7362},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[114,7364],{"className":7365,"style":2032},[2031],[114,7367,7369],{"className":7368},[2036,2037,2038,2039],[114,7370,1964],{"className":7371,"style":2046},[181,182,2039],[114,7373,2052],{"className":7374},[2051],[114,7376,7378],{"className":7377},[2019],[114,7379,7381],{"className":7380,"style":2059},[2023],[114,7382],{},[114,7384,7386,7389],{"className":7385},[181],[114,7387,6970],{"className":7388},[181,182],[114,7390,7392],{"className":7391},[2010],[114,7393,7395,7432],{"className":7394},[2014,2015],[114,7396,7398,7429],{"className":7397},[2019],[114,7399,7402,7414],{"className":7400,"style":7401},[2023],"height:0.8019em;",[114,7403,7405,7408],{"style":7404},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[114,7406],{"className":7407,"style":2032},[2031],[114,7409,7411],{"className":7410},[2036,2037,2038,2039],[114,7412,1964],{"className":7413,"style":2046},[181,182,2039],[114,7415,7417,7420],{"style":7416},"top:-3.113em;margin-right:0.05em;",[114,7418],{"className":7419,"style":2032},[2031],[114,7421,7423],{"className":7422},[2036,2037,2038,2039],[114,7424,7426],{"className":7425},[181,2039],[114,7427,6991],{"className":7428},[181,2039],[114,7430,2052],{"classNam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h43z",[114,7777,2052],{"className":7778},[2051],[114,7780,7782],{"className":7781},[2019],[114,7783,7786],{"className":7784,"style":7785},[2023],"height:1.25em;",[114,7787],{},[114,7789],{"className":7790,"style":250},[191],[114,7792,7794],{"className":7793},[181],[114,7795,7797,7842],{"className":7796},[7069],[114,7798,7801],{"className":7799},[7800],"col-align-r",[114,7802,7804,7833],{"className":7803},[2014,2015],[114,7805,7807,7830],{"className":7806},[2019],[114,7808,7811,7821],{"className":7809,"style":7810},[2023],"height:1.6em;",[114,7812,7814,7818],{"style":7813},"top:-3.6em;",[114,7815],{"className":7816,"style":7817},[2031],"height:2.84em;",[114,7819],{"className":7820},[181],[114,7822,7824,7827],{"style":7823},"top:-2.1em;",[114,7825],{"className":7826,"style":7817},[2031],[114,7828],{"className":7829},[181],[114,7831,2052],{"className":7832},[2051],[114,7834,7836],{"className":7835},[2019],[114,7837,7840],{"className":7838,"style":7839},[2023],"height:1.1em;",[114,7841],{},[114,7843,7846],{"className":7844},[7845],"col-align-l",[114,7847,7849,7951],{"className":7848},[2014,2015],[114,7850,7852,7948],{"className":7851},[2019],[114,7853,7855,7900],{"className":7854,"style":7810},[2023],[114,7856,7858,7861],{"style":7857},"top:-3.76em;",[114,7859],{"className":7860,"style":4525},[2031],[114,7862,7864,7867,7870,7873,7876,7879,7882,7897],{"className":7863},[181],[114,7865],{"className":7866},[181],[114,7868,1956],{"className":7869},[181,182],[114,7871],{"className":7872,"style":250},[191],[114,7874,828],{"className":7875},[254],[114,7877],{"className":7878,"style":250},[191],[114,7880,556],{"className":7881},[181,576],[114,7883,7885,7889,7894],{"className":7884},[181,7222],[114,7886,7888],{"className":7887},[181]," (",[114,7890,7893],{"className":7891},[181,7892],"cjk_fallback","整数倍",[114,7895,155],{"className":7896},[181],[114,7898,144],{"className":7899},[187],[114,7901,7903,7906],{"style":7902},"top:-2.26em;",[114,7904],{"className":7905,"style":4525},[2031],[114,7907,7909,7912,7915,7918,7921,7924,7927,7930,7933,7936,7939,7942,7945],{"className":7908},[181],[114,7910],{"className":7911},[181],[114,7913,6970],{"className":7914},[181,182],[114,7916],{"className":7917,"style":250},[191],[114,7919,828],{"className":7920},[254],[114,7922],{"className":7923,"style":250},[191],[114,7925,1476],{"className":7926,"style":1503},[181,182],[114,7928,144],{"className":7929},[187],[114,7931],{"className":7932,"style":192},[191],[114,7934,825],{"className":7935,"style":856},[181,182],[114,7937],{"className":7938,"style":250},[191],[114,7940,828],{"className":7941},[254],[114,7943],{"className":7944,"style":250},[191],[114,7946,140],{"className":7947,"style":183},[181,182],[114,7949,2052],{"className":7950},[2051],[114,7952,7954],{"className":7953},[2019],[114,7955,7957],{"className":7956,"style":7839},[2023],[114,7958],{},[114,7960,7962],{"className":7961,"style":7186},[208,7185],[114,7963,632],{"className":7964},[7190,7191],[447,7966,7967],{},[450,7968,7969,519,7972],{},[51,7970,7971],{},"解读",[447,7973,7974,8093,8233,8392,8434],{},[450,7975,7976,8092],{},[114,7977,7979,8003],{"className":7978},[117],[114,7980,7982],{"className":7981},[121],[123,7983,7984],{"xmlns":125},[127,7985,7986,8000],{},[130,7987,7988,7994],{},[1952,7989,7990,7992],{},[138,7991,1956],{},[138,7993,1961],{},[1952,7995,7996,7998],{},[138,7997,6970],{},[138,7999,1961],{},[157,8001,8002],{"encoding":159},"a_i m_i",[114,8004,8006],{"className":8005,"ariaHidden":143},[164],[114,8007,8009,8012,8052],{"className":8008},[168],[114,8010],{"className":8011,"style":5667},[172],[114,8013,8015,8018],{"className":8014},[181],[114,8016,1956],{"className":8017},[181,182],[114,8019,8021],{"className":8020},[2010],[114,8022,8024,8044],{"className":8023},[2014,2015],[114,8025,8027,8041],{"className":8026},[2019],[114,8028,8030],{"className":8029,"style":2024},[2023],[114,8031,8032,8035],{"style":2027},[114,8033],{"className":8034,"style":2032},[2031],[114,8036,8038],{"className":8037},[2036,2037,2038,2039],[114,8039,1961],{"className":8040},[181,182,2039],[114,8042,2052],{"className":8043},[2051],[114,8045,8047],{"className":8046},[2019],[114,8048,8050],{"className":8049,"style":2850},[2023],[114,8051],{},[114,8053,8055,8058],{"className":8054},[181],[114,8056,6970],{"className":8057},[181,182],[114,8059,8061],{"className":8060},[2010],[114,8062,8064,8084],{"className":8063},[2014,2015],[114,8065,8067,8081],{"className":8066},[2019],[114,8068,8070],{"className":8069,"style":2024},[2023],[114,8071,8072,8075],{"style":2027},[114,8073],{"className":8074,"style":2032},[2031],[114,8076,8078],{"className":8077},[2036,2037,2038,2039],[114,8079,1961],{"className":8080},[181,182,2039],[114,8082,2052],{"className":8083},[2051],[114,8085,8087],{"className":8086},[2019],[114,8088,8090],{"className":8089,"style":2850},[2023],[114,8091],{},"：保证它是加法子群（整数倍）。",[450,8094,8095,8232],{},[114,8096,8098,8124],{"className":8097},[117],[114,8099,8101],{"className":8100},[121],[123,8102,8103],{"xmlns":125},[127,8104,8105,8121],{},[130,8106,8107,8113],{},[1952,8108,8109,8111],{},[138,8110,825],{},[138,8112,1964],{},[6982,8114,8115,8117,8119],{},[138,8116,6970],{},[138,8118,1964],{},[133,8120,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[157,8122,8123],{"encoding":159},"r_j m_j'",[114,8125,8127],{"className":8126,"ariaHidden":143},[164],[114,8128,8130,8134,8174],{"className":8129},[168],[114,8131],{"className":8132,"style":8133},[172],"height:1.1467em;vertical-align:-0.3948em;",[114,8135,8137,8140],{"className":8136},[181],[114,8138,825],{"className":8139,"style":856},[181,182],[114,8141,8143],{"className":8142},[2010],[114,8144,8146,8166],{"className":8145},[2014,2015],[114,8147,8149,8163],{"className":8148},[2019],[114,8150,8152],{"className":8151,"style":2024},[2023],[114,8153,8154,8157],{"style":7362},[114,8155],{"className":8156,"style":2032},[2031],[114,8158,8160],{"className":8159},[2036,2037,2038,2039],[114,8161,1964],{"className":8162,"style":2046},[181,182,2039],[114,8164,2052],{"className":8165},[2051],[114,8167,8169],{"className":8168},[2019],[114,8170,8172],{"className":8171,"style":2059},[2023],[114,8173],{},[114,8175,8177,8180],{"className":8176},[181],[114,8178,6970],{"className":8179},[181,182],[114,8181,8183],{"className":8182},[2010],[114,8184,8186,8223],{"className":8185},[2014,2015],[114,8187,8189,8220],{"className":8188},[2019],[114,8190,8193,8205],{"className":8191,"style":8192},[2023],"height:0.7519em;",[114,8194,8196,8199],{"style":8195},"top:-2.4413em;margin-left:0em;margin-right:0.05em;",[114,8197],{"className":8198,"style":2032},[2031],[114,8200,8202],{"className":8201},[2036,2037,2038,2039],[114,8203,1964],{"className":8204,"style":2046},[181,182,2039],[114,8206,8208,8211],{"style":8207},"top:-3.063em;margin-right:0.05em;",[114,8209],{"className":8210,"style":2032},[2031],[114,8212,8214],{"className":8213},[2036,2037,2038,2039],[114,8215,8217],{"className":8216},[181,2039],[114,8218,6991],{"className":8219},[181,2039],[114,8221,2052],{"className":8222},[2051],[114,8224,8226],{"className":8225},[2019],[114,8227,8230],{"className":8228,"style":8229},[2023],"height:0.3948em;",[114,8231],{},"：保证左吸收律（左乘）。",[450,8234,8235,8391],{},[114,8236,8238,8270],{"className":8237},[117],[114,8239,8241],{"className":8240},[121],[123,8242,8243],{"xmlns":125},[127,8244,8245,8267],{},[130,8246,8247,8259],{},[6982,8248,8249,8251,8253],{},[138,8250,6970],{},[138,8252,5288],{},[130,8254,8255,8257],{},[133,8256,6991],{"mathvariant":1790},[133,8258,6991],{"mathvariant":1790},[6982,8260,8261,8263,8265],{},[138,8262,825],{},[138,8264,5288],{},[133,8266,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[157,8268,8269],{"encoding":159},"m_k'' r_k'",[114,8271,8273],{"className":8272,"ariaHidden":143},[164],[114,8274,8276,8280,8336],{"className":8275},[168],[114,8277],{"className":8278,"style":8279},[172],"height:1.035em;vertical-align:-0.2831em;",[114,8281,8283,8286],{"className":8282},[181],[114,8284,6970],{"className":8285},[181,182],[114,8287,8289],{"className":8288},[2010],[114,8290,8292,8327],{"className":8291},[2014,2015],[114,8293,8295,8324],{"className":8294},[2019],[114,8296,8298,8310],{"className":8297,"style":8192},[2023],[114,8299,8301,8304],{"style":8300},"top:-2.4169em;margin-left:0em;margin-right:0.05em;",[114,8302],{"className":8303,"style":2032},[2031],[114,8305,8307],{"className":8306},[2036,2037,2038,2039],[114,8308,5288],{"className":8309,"style":5304},[181,182,2039],[114,8311,8312,8315],{"style":8207},[114,8313],{"className":8314,"style":2032},[2031],[114,8316,8318],{"className":8317},[2036,2037,2038,2039],[114,8319,8321],{"className":8320},[181,2039],[114,8322,7493],{"className":8323},[181,2039],[114,8325,2052],{"className":8326},[2051],[114,8328,8330],{"className":8329},[2019],[114,8331,8334],{"className":8332,"style":8333},[2023],"height:0.2831em;",[114,8335],{},[114,8337,8339,8342],{"className":8338},[181],[114,8340,825],{"className":8341,"style":856},[181,182],[114,8343,8345],{"className":8344},[2010],[114,8346,8348,8383],{"className":8347},[2014,2015],[114,8349,8351,8380],{"className":8350},[2019],[114,8352,8354,8366],{"className":8353,"style":8192},[2023],[114,8355,8357,8360],{"style":8356},"top:-2.4169em;margin-left:-0.0278em;margin-right:0.05em;",[114,8358],{"className":8359,"style":2032},[2031],[114,8361,8363],{"className":8362},[2036,2037,2038,2039],[114,8364,5288],{"className":8365,"style":5304},[181,182,2039],[114,8367,8368,8371],{"style":8207},[114,8369],{"className":8370,"style":2032},[2031],[114,8372,8374],{"className":8373},[2036,2037,2038,2039],[114,8375,8377],{"className":8376},[181,2039],[114,8378,6991],{"className":8379},[181,2039],[114,8381,2052],{"className":8382},[2051],[114,8384,8386],{"className":8385},[2019],[114,8387,8389],{"className":8388,"style":8333},[2023],[114,8390],{},"：保证右吸收律（右乘）。",[450,8393,8394,8433],{},[114,8395,8397,8415],{"className":8396},[117],[114,8398,8400],{"className":8399},[121],[123,8401,8402],{"xmlns":125},[127,8403,8404,8412],{},[130,8405,8406,8408,8410],{},[138,8407,825],{},[138,8409,6970],{},[138,8411,825],{},[157,8413,8414],{"encoding":159},"r m r",[114,8416,8418],{"className":8417,"ariaHidden":143},[164],[114,8419,8421,8424,8427,8430],{"className":8420},[168],[114,8422],{"className":8423,"style":1420},[172],[114,8425,825],{"className":8426,"style":856},[181,182],[114,8428,6970],{"className":8429},[181,182],[114,8431,825],{"className":8432,"style":856},[181,182],"：保证双侧乘积。",[450,8435,8436,8437,8465],{},"这四种形式的有限和，是包含 ",[114,8438,8440,8453],{"className":8439},[117],[114,8441,8443],{"className":8442},[121],[123,8444,8445],{"xmlns":125},[127,8446,8447,8451],{},[130,8448,8449],{},[138,8450,1476],{},[157,8452,1476],{"encoding":159},[114,8454,8456],{"className":8455,"ariaHidden":143},[164],[114,8457,8459,8462],{"className":8458},[168],[114,8460],{"className":8461,"style":264},[172],[114,8463,1476],{"className":8464,"style":1503},[181,182]," 且满足理想定义的最小集合。",[19,8467,8468,8471,8472,8533],{},[51,8469,8470],{},"情况 2：交换环 (Commutative Ring)","\n由于 ",[114,8473,8475,8497],{"className":8474},[117],[114,8476,8478],{"className":8477},[121],[123,8479,8480],{"xmlns":125},[127,8481,8482,8494],{},[130,8483,8484,8486,8488,8490,8492],{},[138,8485,825],{},[138,8487,6970],{},[133,8489,1403],{},[138,8491,6970],{},[138,8493,825],{},[157,8495,8496],{"encoding":159},"rm = mr",[114,8498,8500,8521],{"className":8499,"ariaHidden":143},[164],[114,8501,8503,8506,8509,8512,8515,8518],{"className":8502},[168],[114,8504],{"className":8505,"style":1420},[172],[114,8507,825],{"className":8508,"style":856},[181,182],[114,8510,6970],{"className":8511},[181,182],[114,8513],{"className":8514,"style":250},[191],[114,8516,1403],{"className":8517},[254],[114,8519],{"className":8520,"style":250},[191],[114,8522,8524,8527,8530],{"className":8523},[168],[114,8525],{"className":8526,"style":1420},[172],[114,8528,6970],{"className":8529},[181,182],[114,8531,825],{"className":8532,"style":856},[181,182],"，左右乘不分家。",[114,8535,8537],{"className":8536},[6923],[114,8538,8540,8629],{"className":8539},[117],[114,8541,8543],{"className":8542},[121],[123,8544,8545],{"xmlns":125,"display":6932},[127,8546,8547,8626],{},[130,8548,8549,8551,8553,8555,8557,8618,8620,8622,8624],{},[133,8550,6478],{},[138,8552,1476],{},[133,8554,6483],{},[133,8556,1403],{},[130,8558,8559,8561,8567,8573,8579,8587,8589,8591,8593,8595,8597,8604,8606,8608,8610,8612,8614,8616],{},[133,8560,626],{"fence":143},[6950,8562,8563,8565],{},[133,8564,5631],{},[6955,8566,6957],{},[1952,8568,8569,8571],{},[138,8570,825],{},[138,8572,1961],{},[1952,8574,8575,8577],{},[138,8576,6970],{},[138,8578,1961],{},[6982,8580,8581,8583,8585],{},[138,8582,825],{},[138,8584,1961],{},[133,8586,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[6955,8588,7061],{},[133,8590,7065],{"fence":143,"lspace":7064,"rspace":7064},[6955,8592,7061],{},[138,8594,825],{},[133,8596,144],{"separator":143},[8598,8599,8600,8602],"msup",{},[138,8601,825],{},[133,8603,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[133,8605,828],{},[138,8607,140],{},[133,8609,144],{"separator":143},[138,8611,6970],{},[133,8613,828],{},[138,8615,1476],{},[133,8617,632],{"fence":143},[133,8619,1403],{},[138,8621,140],{},[138,8623,1476],{},[138,8625,140],{},[157,8627,8628],{"encoding":159},"\u003C M > = \\left\\{ \\sum_{\\text{finite}} r_i m_i r_i' \\;\\middle|\\; r, r' \\in R, m \\in M \\right\\} = RMR",[114,8630,8632,8644,8662,8995],{"className":8631,"ariaHidden":143},[164],[114,8633,8635,8638,8641],{"className":8634},[168],[114,8636],{"className":8637,"style":852},[172],[114,8639,6478],{"className":8640},[254],[114,8642],{"className":8643,"style":250},[191],[114,8645,8647,8650,8653,8656,8659],{"className":8646},[168],[114,8648],{"className":8649,"style":1910},[172],[114,8651,1476],{"className":8652,"style":1503},[181,182],[114,8654],{"className":8655,"style":250},[191],[114,8657,6707],{"className":8658},[254],[114,8660],{"className":8661,"style":250},[191],[114,8663,8665,8668,8986,8989,8992],{"className":8664},[168],[114,8666],{"className":8667,"style":7177},[172],[114,8669,8671,8677,8727,8730,8770,8810,8864,8867,8903,8906,8909,8912,8915,8947,8950,8953,8956,8959,8962,8965,8968,8971,8974,8977,8980],{"className":8670},[7181],[114,8672,8674],{"className":8673,"style":7186},[177,7185],[114,8675,626],{"className":8676},[7190,7191],[114,8678,8680],{"className":8679},[5732,7195],[114,8681,8683,8719],{"className":8682},[2014,2015],[114,8684,8686,8716],{"className":8685},[2019],[114,8687,8689,8706],{"className":8688,"style":7205},[2023],[114,8690,8691,8694],{"style":7208},[114,8692],{"className":8693,"style":7212},[2031],[114,8695,8697],{"className":8696},[2036,2037,2038,2039],[114,8698,8700],{"className":8699},[181,2039],[114,8701,8703],{"className":8702},[181,7222,2039],[114,8704,6957],{"className":8705},[181,2039],[114,8707,8708,8711],{"style":7228},[114,8709],{"className":8710,"style":7212},[2031],[114,8712,8713],{},[114,8714,5631],{"className":8715},[5732,5736,7237],[114,8717,2052],{"className":8718},[2051],[114,8720,8722],{"className":8721},[2019],[114,8723,8725],{"className":8724,"style":7247},[2023],[114,8726],{},[114,8728],{"className":8729,"style":192},[191],[114,8731,8733,8736],{"className":8732},[181],[114,8734,825],{"className":8735,"style":856},[181,182],[114,8737,8739],{"className":8738},[2010],[114,8740,8742,8762],{"className":8741},[2014,2015],[114,8743,8745,8759],{"className":8744},[2019],[114,8746,8748],{"className":8747,"style":2024},[2023],[114,8749,8750,8753],{"style":7362},[114,8751],{"className":8752,"style":2032},[2031],[114,8754,8756],{"className":8755},[2036,2037,2038,2039],[114,8757,1961],{"className":8758},[181,182,2039],[114,8760,2052],{"className":8761},[2051],[114,8763,8765],{"className":8764},[2019],[114,8766,8768],{"className":8767,"style":2850},[2023],[114,8769],{},[114,8771,8773,8776],{"className":8772},[181],[114,8774,6970],{"className":8775},[181,182],[114,8777,8779],{"className":8778},[2010],[114,8780,8782,8802],{"className":8781},[2014,2015],[114,8783,8785,8799],{"className":8784},[2019],[114,8786,8788],{"className":8787,"style":2024},[2023],[114,8789,8790,8793],{"style":2027},[114,8791],{"className":8792,"style":2032},[2031],[114,8794,8796],{"className":8795},[2036,2037,2038,2039],[114,8797,1961],{"className":8798},[181,182,2039],[114,8800,2052],{"className":8801},[2051],[114,8803,8805],{"className":8804},[2019],[114,8806,8808],{"className":8807,"style":2850},[2023],[114,8809],{},[114,8811,8813,8816],{"className":8812},[181],[114,8814,825],{"className":8815,"style":856},[181,182],[114,8817,8819],{"className":8818},[2010],[114,8820,8822,8856],{"className":8821},[2014,2015],[114,8823,8825,8853],{"className":8824},[2019],[114,8826,8828,8839],{"className":8827,"style":7401},[2023],[114,8829,8830,8833],{"style":7526},[114,8831],{"className":8832,"style":2032},[2031],[114,8834,8836],{"className":8835},[2036,2037,2038,2039],[114,8837,1961],{"className":8838},[181,182,2039],[114,8840,8841,8844],{"style":7416},[114,8842],{"className":8843,"style":2032},[2031],[114,8845,8847],{"className":8846},[2036,2037,2038,2039],[114,8848,8850],{"className":8849},[181,2039],[114,8851,6991],{"className":8852},[181,2039],[114,8854,2052],{"className":8855},[2051],[114,8857,8859],{"className":8858},[2019],[114,8860,8862],{"className":8861,"style":7503},[2023],[114,8863],{},[114,8865],{"className":8866,"style":250},[191],[114,8868,8869],{},[114,8870,8872],{"className":8871},[7190,7744],[114,8873,8875,8895],{"className":8874},[2014,2015],[114,8876,8878,8892],{"className":8877},[2019],[114,8879,8881],{"className":8880,"style":7754},[2023],[114,8882,8883,8886],{"style":7757},[114,8884],{"className":8885,"style":7761},[2031],[114,8887,8888],{"style":7764},[7766,8889,8890],{"xmlns":7768,"width":7769,"height":7770,"viewBox":7771},[7773,8891],{"d":7775},[114,8893,2052],{"className":8894},[2051],[114,8896,8898],{"className":8897},[2019],[114,8899,8901],{"className":8900,"style":7785},[2023],[114,8902],{},[114,8904],{"className":8905,"style":250},[191],[114,8907,825],{"className":8908,"style":856},[181,182],[114,8910,144],{"className":8911},[187],[114,8913],{"className":8914,"style":192},[191],[114,8916,8918,8921],{"className":8917},[181],[114,8919,825],{"className":8920,"style":856},[181,182],[114,8922,8924],{"className":8923},[2010],[114,8925,8927],{"className":8926},[2014],[114,8928,8930],{"className":8929},[2019],[114,8931,8933],{"className":8932,"style":7401},[2023],[114,8934,8935,8938],{"style":7416},[114,8936],{"className":8937,"style":2032},[2031],[114,8939,8941],{"className":8940},[2036,2037,2038,2039],[114,8942,8944],{"className":8943},[181,2039],[114,8945,6991],{"className":8946},[181,2039],[114,8948],{"className":8949,"style":250},[191],[114,8951,828],{"className":8952},[254],[114,8954],{"className":8955,"style":250},[191],[114,8957,140],{"className":8958,"style":183},[181,182],[114,8960,144],{"className":8961},[187],[114,8963],{"className":8964,"style":192},[191],[114,8966,6970],{"className":8967},[181,182],[114,8969],{"className":8970,"style":250},[191],[114,8972,828],{"className":8973},[254],[114,8975],{"className":8976,"style":250},[191],[114,8978,1476],{"className":8979,"style":1503},[181,182],[114,8981,8983],{"className":8982,"style":7186},[208,7185],[114,8984,632],{"className":8985},[7190,7191],[114,8987],{"className":8988,"style":250},[191],[114,8990,1403],{"className":8991},[254],[114,8993],{"className":8994,"style":250},[191],[114,8996,8998,9001,9004,9007],{"className":8997},[168],[114,8999],{"className":9000,"style":264},[172],[114,9002,140],{"className":9003,"style":183},[181,182],[114,9005,1476],{"className":9006,"style":1503},[181,182],[114,9008,140],{"className":9009,"style":183},[181,182],[19,9011,9012],{},[533,9013,9014,9015,155],{},"(注：如果环没有单位元，可能还需要加上整数倍项 ",[114,9016,9018,9034],{"className":9017},[117],[114,9019,9021],{"className":9020},[121],[123,9022,9023],{"xmlns":125},[127,9024,9025,9031],{},[130,9026,9027,9029],{},[138,9028,1481],{},[138,9030,6970],{},[157,9032,9033],{"encoding":159},"n m",[114,9035,9037],{"className":9036,"ariaHidden":143},[164],[114,9038,9040,9043],{"className":9039},[168],[114,9041],{"className":9042,"style":1420},[172],[114,9044,9046],{"className":9045},[181,182],"nm",[19,9048,9049,9052,9053,9081,9082,9112,9113,9179,9180,9214],{},[51,9050,9051],{},"情况 3：有单位元的交换环 (Commutative Ring with Unity)","\n这是最常见的情况。因为有单位元 ",[114,9054,9056,9069],{"className":9055},[117],[114,9057,9059],{"className":9058},[121],[123,9060,9061],{"xmlns":125},[127,9062,9063,9067],{},[130,9064,9065],{},[550,9066,4373],{},[157,9068,4373],{"encoding":159},[114,9070,9072],{"className":9071,"ariaHidden":143},[164],[114,9073,9075,9078],{"className":9074},[168],[114,9076],{"className":9077,"style":2149},[172],[114,9079,4373],{"className":9080},[181],"，整数倍 ",[114,9083,9085,9100],{"className":9084},[117],[114,9086,9088],{"className":9087},[121],[123,9089,9090],{"xmlns":125},[127,9091,9092,9098],{},[130,9093,9094,9096],{},[138,9095,1481],{},[138,9097,6970],{},[157,9099,9033],{"encoding":159},[114,9101,9103],{"className":9102,"ariaHidden":143},[164],[114,9104,9106,9109],{"className":9105},[168],[114,9107],{"className":9108,"style":1420},[172],[114,9110,9046],{"className":9111},[181,182]," 可以写成 ",[114,9114,9116,9140],{"className":9115},[117],[114,9117,9119],{"className":9118},[121],[123,9120,9121],{"xmlns":125},[127,9122,9123,9137],{},[130,9124,9125,9127,9129,9131,9133,9135],{},[133,9126,136],{"stretchy":135},[138,9128,1481],{},[133,9130,152],{},[550,9132,4373],{},[133,9134,155],{"stretchy":135},[138,9136,6970],{},[157,9138,9139],{"encoding":159},"(n \\cdot 1)m",[114,9141,9143,9164],{"className":9142,"ariaHidden":143},[164],[114,9144,9146,9149,9152,9155,9158,9161],{"className":9145},[168],[114,9147],{"className":9148,"style":173},[172],[114,9150,136],{"className":9151},[177],[114,9153,1481],{"className":9154},[181,182],[114,9156],{"className":9157,"style":652},[191],[114,9159,152],{"className":9160},[656],[114,9162],{"className":9163,"style":652},[191],[114,9165,9167,9170,9173,9176],{"className":9166},[168],[114,9168],{"className":9169,"style":173},[172],[114,9171,4373],{"className":9172},[181],[114,9174,155],{"className":9175},[208],[114,9177,6970],{"className":9178},[181,182],"，被 ",[114,9181,9183,9199],{"className":9182},[117],[114,9184,9186],{"className":9185},[121],[123,9187,9188],{"xmlns":125},[127,9189,9190,9196],{},[130,9191,9192,9194],{},[138,9193,825],{},[138,9195,6970],{},[157,9197,9198],{"encoding":159},"rm",[114,9200,9202],{"className":9201,"ariaHidden":143},[164],[114,9203,9205,9208,9211],{"className":9204},[168],[114,9206],{"className":9207,"style":1420},[172],[114,9209,825],{"className":9210,"style":856},[181,182],[114,9212,6970],{"className":9213},[181,182]," 形式覆盖。",[114,9216,9218],{"className":9217},[6923],[114,9219,9221,9299],{"className":9220},[117],[114,9222,9224],{"className":9223},[121],[123,9225,9226],{"xmlns":125,"display":6932},[127,9227,9228,9296],{},[130,9229,9230,9232,9234,9236,9238,9290,9292,9294],{},[133,9231,6478],{},[138,9233,1476],{},[133,9235,6483],{},[133,9237,1403],{},[130,9239,9240,9242,9248,9254,9260,9262,9264,9266,9272,9274,9276,9278,9284,9286,9288],{},[133,9241,626],{"fence":143},[6950,9243,9244,9246],{},[133,9245,5631],{},[6955,9247,6957],{},[1952,9249,9250,9252],{},[138,9251,825],{},[138,9253,1961],{},[1952,9255,9256,9258],{},[138,9257,6970],{},[138,9259,1961],{},[6955,9261,7061],{},[133,9263,7065],{"fence":143,"lspace":7064,"rspace":7064},[6955,9265,7061],{},[1952,9267,9268,9270],{},[138,9269,825],{},[138,9271,1961],{},[133,9273,828],{},[138,9275,140],{},[133,9277,144],{"separator":143},[1952,9279,9280,9282],{},[138,9281,6970],{},[138,9283,1961],{},[133,9285,828],{},[138,9287,1476],{},[133,9289,632],{"fence":143},[133,9291,1403],{},[138,9293,140],{},[138,9295,1476],{},[157,9297,9298],{"encoding":159},"\u003C M > = \\left\\{ \\sum_{\\text{finite}} r_i m_i \\;\\middle|\\; r_i \\in R, m_i \\in M \\right\\} = RM",[114,9300,9302,9314,9332,9647],{"className":9301,"ariaHidden":143},[164],[114,9303,9305,9308,9311],{"className":9304},[168],[114,9306],{"className":9307,"style":852},[172],[114,9309,6478],{"className":9310},[254],[114,9312],{"className":9313,"style":250},[191],[114,9315,9317,9320,9323,9326,9329],{"className":9316},[168],[114,9318],{"className":9319,"style":1910},[172],[114,9321,1476],{"className":9322,"style":1503},[181,182],[114,9324],{"className":9325,"style":250},[191],[114,9327,6707],{"className":9328},[254],[114,9330],{"className":9331,"style":250},[191],[114,9333,9335,9338,9638,9641,9644],{"className":9334},[168],[114,9336],{"className":9337,"style":7177},[172],[114,9339,9341,9347,9397,9400,9440,9480,9483,9519,9522,9562,9565,9568,9571,9574,9577,9580,9620,9623,9626,9629,9632],{"className":9340},[7181],[114,9342,9344],{"className":9343,"style":7186},[177,7185],[114,9345,626],{"className":9346},[7190,7191],[114,9348,9350],{"className":9349},[5732,7195],[114,9351,9353,9389],{"className":9352},[2014,2015],[114,9354,9356,9386],{"className":9355},[2019],[114,9357,9359,9376],{"className":9358,"style":7205},[2023],[114,9360,9361,9364],{"style":7208},[114,9362],{"className":9363,"style":7212},[2031],[114,9365,9367],{"className":9366},[2036,2037,2038,2039],[114,9368,9370],{"className":9369},[181,2039],[114,9371,9373],{"className":9372},[181,7222,2039],[114,9374,6957],{"className":9375},[181,2039],[114,9377,9378,9381],{"style":7228},[114,9379],{"className":9380,"style":7212},[2031],[114,9382,9383],{},[114,9384,5631],{"className":9385},[5732,5736,7237],[114,9387,2052],{"className":9388},[2051],[114,9390,9392],{"className":9391},[2019],[114,9393,9395],{"className":9394,"style":7247},[2023],[114,9396],{},[114,9398],{"className":9399,"style":192},[191],[114,9401,9403,9406],{"className":9402},[181],[114,9404,825],{"className":9405,"style":856},[181,182],[114,9407,9409],{"className":9408},[2010],[114,9410,9412,9432],{"className":9411},[2014,2015],[114,9413,9415,9429],{"className":9414},[2019],[114,9416,9418],{"className":9417,"style":2024},[2023],[114,9419,9420,9423],{"style":7362},[114,9421],{"className":9422,"style":2032},[2031],[114,9424,9426],{"className":9425},[2036,2037,2038,2039],[114,9427,1961],{"className":9428},[181,182,2039],[114,9430,2052],{"className":9431},[2051],[114,9433,9435],{"className":9434},[2019],[114,9436,9438],{"className":9437,"style":2850},[2023],[114,9439],{},[114,9441,9443,9446],{"className":9442},[181],[114,9444,6970],{"className":9445},[181,182],[114,9447,9449],{"className":9448},[2010],[114,9450,9452,9472],{"className":9451},[2014,2015],[114,9453,9455,9469],{"className":9454},[2019],[114,9456,9458],{"className":9457,"style":2024},[2023],[114,9459,9460,9463],{"style":2027},[114,9461],{"className":9462,"style":2032},[2031],[114,9464,9466],{"className":9465},[2036,2037,2038,2039],[114,9467,1961],{"className":9468},[181,182,2039],[114,9470,2052],{"className":9471},[2051],[114,9473,9475],{"className":9474},[2019],[114,9476,9478],{"className":9477,"style":2850},[2023],[114,9479],{},[114,9481],{"className":9482,"style":250},[191],[114,9484,9485],{},[114,9486,9488],{"className":9487},[7190,7744],[114,9489,9491,9511],{"className":9490},[2014,2015],[114,9492,9494,9508],{"className":9493},[2019],[114,9495,9497],{"className":9496,"style":7754},[2023],[114,9498,9499,9502],{"style":7757},[114,9500],{"className":9501,"style":7761},[2031],[114,9503,9504],{"style":7764},[7766,9505,9506],{"xmlns":7768,"width":7769,"height":7770,"viewBox":7771},[7773,9507],{"d":7775},[114,9509,2052],{"className":9510},[2051],[114,9512,9514],{"className":9513},[2019],[114,9515,9517],{"className":9516,"style":7785},[2023],[114,9518],{},[114,9520],{"className":9521,"style":250},[191],[114,9523,9525,9528],{"className":9524},[181],[114,9526,825],{"className":9527,"style":856},[181,182],[114,9529,9531],{"className":9530},[2010],[114,9532,9534,9554],{"className":9533},[2014,2015],[114,9535,9537,9551],{"className":9536},[2019],[114,9538,9540],{"className":9539,"style":2024},[2023],[114,9541,9542,9545],{"style":7362},[114,9543],{"className":9544,"style":2032},[2031],[114,9546,9548],{"className":9547},[2036,2037,2038,2039],[114,9549,1961],{"className":9550},[181,182,2039],[114,9552,2052],{"className":9553},[2051],[114,9555,9557],{"className":9556},[2019],[114,9558,9560],{"className":9559,"style":2850},[2023],[114,9561],{},[114,9563],{"className":9564,"style":250},[191],[114,9566,828],{"className":9567},[254],[114,9569],{"className":9570,"style":250},[191],[114,9572,140],{"className":9573,"style":183},[181,182],[114,9575,144],{"className":9576},[187],[114,9578],{"className":9579,"style":192},[191],[114,9581,9583,9586],{"className":9582},[181],[114,9584,6970],{"className":9585},[181,182],[114,9587,9589],{"className":9588},[2010],[114,9590,9592,9612],{"className":9591},[2014,2015],[114,9593,9595,9609],{"className":9594},[2019],[114,9596,9598],{"className":9597,"style":2024},[2023],[114,9599,9600,9603],{"style":2027},[114,9601],{"className":9602,"style":2032},[2031],[114,9604,9606],{"className":9605},[2036,2037,2038,2039],[114,9607,1961],{"className":9608},[181,182,2039],[114,9610,2052],{"className":9611},[2051],[114,9613,9615],{"className":9614},[2019],[114,9616,9618],{"className":9617,"style":2850},[2023],[114,9619],{},[114,9621],{"className":9622,"style":250},[191],[114,9624,828],{"className":9625},[254],[114,9627],{"className":9628,"style":250},[191],[114,9630,1476],{"className":9631,"style":1503},[181,182],[114,9633,9635],{"className":9634,"style":7186},[208,7185],[114,9636,632],{"className":9637},[7190,7191],[114,9639],{"className":9640,"style":250},[191],[114,9642,1403],{"className":9643},[254],[114,9645],{"className":9646,"style":250},[191],[114,9648,9650,9653,9656],{"className":9649},[168],[114,9651],{"className":9652,"style":264},[172],[114,9654,140],{"className":9655,"style":183},[181,182],[114,9657,1476],{"className":9658,"style":1503},[181,182],[19,9660,9661,9662,9713,9714,445],{},"特别地，对于主理想 ",[114,9663,9665,9683],{"className":9664},[117],[114,9666,9668],{"className":9667},[121],[123,9669,9670],{"xmlns":125},[127,9671,9672,9680],{},[130,9673,9674,9676,9678],{},[133,9675,6478],{},[138,9677,1956],{},[133,9679,6483],{},[157,9681,9682],{"encoding":159},"\u003C a >",[114,9684,9686,9698],{"className":9685,"ariaHidden":143},[164],[114,9687,9689,9692,9695],{"className":9688},[168],[114,9690],{"className":9691,"style":852},[172],[114,9693,6478],{"className":9694},[254],[114,9696],{"className":9697,"style":250},[191],[114,9699,9701,9704,9707,9710],{"className":9700},[168],[114,9702],{"className":9703,"style":852},[172],[114,9705,1956],{"className":9706},[181,182],[114,9708],{"className":9709,"style":250},[191],[114,9711,6483],{"className":9712},[254],"，就是 ",[114,9715,9717,9752],{"className":9716},[117],[114,9718,9720],{"className":9719},[121],[123,9721,9722],{"xmlns":125},[127,9723,9724,9749],{},[130,9725,9726,9728,9730,9732,9734,9736,9738,9741,9743,9745,9747],{},[138,9727,140],{},[138,9729,1956],{},[133,9731,1403],{},[133,9733,626],{"stretchy":135},[138,9735,825],{},[138,9737,1956],{},[133,9739,9740],{},"∣",[138,9742,825],{},[133,9744,828],{},[138,9746,140],{},[133,9748,632],{"stretchy":135},[157,9750,9751],{"encoding":159},"Ra = \\{ra \\mid r \\in R\\}",[114,9753,9755,9776,9800,9818],{"className":9754,"ariaHidden":143},[164],[114,9756,9758,9761,9764,9767,9770,9773],{"className":9757},[168],[114,9759],{"className":9760,"style":264},[172],[114,9762,140],{"className":9763,"style":183},[181,182],[114,9765,1956],{"className":9766},[181,182],[114,9768],{"className":9769,"style":250},[191],[114,9771,1403],{"className":9772},[254],[114,9774],{"className":9775,"style":250},[191],[114,9777,9779,9782,9785,9788,9791,9794,9797],{"className":9778},[168],[114,9780],{"className":9781,"style":173},[172],[114,9783,626],{"className":9784},[177],[114,9786,825],{"className":9787,"style":856},[181,182],[114,9789,1956],{"className":9790},[181,182],[114,9792],{"className":9793,"style":250},[191],[114,9795,9740],{"className":9796},[254],[114,9798],{"className":9799,"style":250},[191],[114,9801,9803,9806,9809,9812,9815],{"className":9802},[168],[114,9804],{"className":9805,"style":852},[172],[114,9807,825],{"className":9808,"style":856},[181,182],[114,9810],{"className":9811,"style":250},[191],[114,9813,828],{"className":9814},[254],[114,9816],{"className":9817,"style":250},[191],[114,9819,9821,9824,9827],{"className":9820},[168],[114,9822],{"className":9823,"style":173},[172],[114,9825,140],{"className":9826,"style":183},[181,182],[114,9828,632],{"className":9829},[208],[99,9831],{},[14,9833,9835],{"id":9834},"_5-理想的运算","5. 理想的运算",[19,9837,112,9838,772,9882,9910],{},[114,9839,9841,9860],{"className":9840},[117],[114,9842,9844],{"className":9843},[121],[123,9845,9846],{"xmlns":125},[127,9847,9848,9857],{},[130,9849,9850,9852,9854],{},[138,9851,756],{},[133,9853,144],{"separator":143},[138,9855,9856],{},"J",[157,9858,9859],{"encoding":159},"I, J",[114,9861,9863],{"className":9862,"ariaHidden":143},[164],[114,9864,9866,9869,9872,9875,9878],{"className":9865},[168],[114,9867],{"className":9868,"style":872},[172],[114,9870,756],{"className":9871,"style":771},[181,182],[114,9873,144],{"className":9874},[187],[114,9876],{"className":9877,"style":192},[191],[114,9879,9856],{"className":9880,"style":9881},[181,182],"margin-right:0.0962em;",[114,9883,9885,9898],{"className":9884},[117],[114,9886,9888],{"className":9887},[121],[123,9889,9890],{"xmlns":125},[127,9891,9892,9896],{},[130,9893,9894],{},[138,9895,140],{},[157,9897,140],{"encoding":159},[114,9899,9901],{"className":9900,"ariaHidden":143},[164],[114,9902,9904,9907],{"className":9903},[168],[114,9905],{"className":9906,"style":264},[172],[114,9908,140],{"className":9909,"style":183},[181,182]," 的两个理想。",[106,9912,9914,9915],{"id":9913},"_51-交集-iji-cap-jij","5.1 交集 ",[114,9916,9918,9937],{"className":9917},[117],[114,9919,9921],{"className":9920},[121],[123,9922,9923],{"xmlns":125},[127,9924,9925,9934],{},[130,9926,9927,9929,9932],{},[138,9928,756],{},[133,9930,9931],{},"∩",[138,9933,9856],{},[157,9935,9936],{"encoding":159},"I \\cap J",[114,9938,9940,9958],{"className":9939,"ariaHidden":143},[164],[114,9941,9943,9946,9949,9952,9955],{"className":9942},[168],[114,9944],{"className":9945,"style":264},[172],[114,9947,756],{"className":9948,"style":771},[181,182],[114,9950],{"className":9951,"style":652},[191],[114,9953,9931],{"className":9954},[656],[114,9956],{"className":9957,"style":652},[191],[114,9959,9961,9964],{"className":9960},[168],[114,9962],{"className":9963,"style":264},[172],[114,9965,9856],{"className":9966,"style":9881},[181,182],[19,9968,9969,536,9972,10022,10023,536],{},[51,9970,9971],{},"命题",[114,9973,9975,9992],{"className":9974},[117],[114,9976,9978],{"className":9977},[121],[123,9979,9980],{"xmlns":125},[127,9981,9982,9990],{},[130,9983,9984,9986,9988],{},[138,9985,756],{},[133,9987,9931],{},[138,9989,9856],{},[157,9991,9936],{"encoding":159},[114,9993,9995,10013],{"className":9994,"ariaHidden":143},[164],[114,9996,9998,10001,10004,10007,10010],{"className":9997},[168],[114,9999],{"className":10000,"style":264},[172],[114,10002,756],{"className":10003,"style":771},[181,182],[114,10005],{"className":10006,"style":652},[191],[114,10008,9931],{"className":10009},[656],[114,10011],{"className":10012,"style":652},[191],[114,10014,10016,10019],{"className":10015},[168],[114,10017],{"className":10018,"style":264},[172],[114,10020,9856],{"className":10021,"style":9881},[181,182]," 是一个理想。\n",[51,10024,10025],{},"证明",[803,10027,10028,10536],{},[450,10029,10030,1355,10033,10119,10120,10184,10185,10249,10250,10291,10292,10184,10368,5397,10441,445],{},[51,10031,10032],{},"子群判定",[114,10034,10036,10062],{"className":10035},[117],[114,10037,10039],{"className":10038},[121],[123,10040,10041],{"xmlns":125},[127,10042,10043,10059],{},[130,10044,10045,10047,10049,10051,10053,10055,10057],{},[138,10046,1956],{},[133,10048,144],{"separator":143},[138,10050,5648],{},[133,10052,828],{},[138,10054,756],{},[133,10056,9931],{},[138,10058,9856],{},[157,10060,10061],{"encoding":159},"a, b \\in I \\cap J",[114,10063,10065,10092,10110],{"className":10064,"ariaHidden":143},[164],[114,10066,10068,10071,10074,10077,10080,10083,10086,10089],{"className":10067},[168],[114,10069],{"className":10070,"style":1810},[172],[114,10072,1956],{"className":10073},[181,182],[114,10075,144],{"className":10076},[187],[114,10078],{"className":10079,"style":192},[191],[114,10081,5648],{"className":10082},[181,182],[114,10084],{"className":10085,"style":250},[191],[114,10087,828],{"className":10088},[254],[114,10090],{"className":10091,"style":250},[191],[114,10093,10095,10098,10101,10104,10107],{"className":10094},[168],[114,10096],{"className":10097,"style":264},[172],[114,10099,756],{"className":10100,"style":771},[181,182],[114,10102],{"className":10103,"style":652},[191],[114,10105,9931],{"className":10106},[656],[114,10108],{"className":10109,"style":652},[191],[114,10111,10113,10116],{"className":10112},[168],[114,10114],{"className":10115,"style":264},[172],[114,10117,9856],{"className":10118,"style":9881},[181,182],"，则 ",[114,10121,10123,10145],{"className":10122},[117],[114,10124,10126],{"className":10125},[121],[123,10127,10128],{"xmlns":125},[127,10129,10130,10142],{},[130,10131,10132,10134,10136,10138,10140],{},[138,10133,1956],{},[133,10135,144],{"separator":143},[138,10137,5648],{},[133,10139,828],{},[138,10141,756],{},[157,10143,10144],{"encoding":159},"a, b \\in I",[114,10146,10148,10175],{"className":10147,"ariaHidden":143},[164],[114,10149,10151,10154,10157,10160,10163,10166,10169,10172],{"className":10150},[168],[114,10152],{"className":10153,"style":1810},[172],[114,10155,1956],{"className":10156},[181,182],[114,10158,144],{"className":10159},[187],[114,10161],{"className":10162,"style":192},[191],[114,10164,5648],{"className":10165},[181,182],[114,10167],{"className":10168,"style":250},[191],[114,10170,828],{"className":10171},[254],[114,10173],{"className":10174,"style":250},[191],[114,10176,10178,10181],{"className":10177},[168],[114,10179],{"className":10180,"style":264},[172],[114,10182,756],{"className":10183,"style":771},[181,182]," 且 ",[114,10186,10188,10210],{"className":10187},[117],[114,10189,10191],{"className":10190},[121],[123,10192,10193],{"xmlns":125},[127,10194,10195,10207],{},[130,10196,10197,10199,10201,10203,10205],{},[138,10198,1956],{},[133,10200,144],{"separator":143},[138,10202,5648],{},[133,10204,828],{},[138,10206,9856],{},[157,10208,10209],{"encoding":159},"a, b \\in J",[114,10211,10213,10240],{"className":10212,"ariaHidden":143},[164],[114,10214,10216,10219,10222,10225,10228,10231,10234,10237],{"className":10215},[168],[114,10217],{"className":10218,"style":1810},[172],[114,10220,1956],{"className":10221},[181,182],[114,10223,144],{"className":10224},[187],[114,10226],{"className":10227,"style":192},[191],[114,10229,5648],{"className":10230},[181,182],[114,10232],{"className":10233,"style":250},[191],[114,10235,828],{"className":10236},[254],[114,10238],{"className":10239,"style":250},[191],[114,10241,10243,10246],{"className":10242},[168],[114,10244],{"className":10245,"style":264},[172],[114,10247,9856],{"className":10248,"style":9881},[181,182],"。因 ",[114,10251,10253,10270],{"className":10252},[117],[114,10254,10256],{"className":10255},[121],[123,10257,10258],{"xmlns":125},[127,10259,10260,10268],{},[130,10261,10262,10264,10266],{},[138,10263,756],{},[133,10265,144],{"separator":143},[138,10267,9856],{},[157,10269,9859],{"encoding":159},[114,10271,10273],{"className":10272,"ariaHidden":143},[164],[114,10274,10276,10279,10282,10285,10288],{"className":10275},[168],[114,10277],{"className":10278,"style":872},[172],[114,10280,756],{"className":10281,"style":771},[181,182],[114,10283,144],{"className":10284},[187],[114,10286],{"className":10287,"style":192},[191],[114,10289,9856],{"className":10290,"style":9881},[181,182]," 是加法群，故 ",[114,10293,10295,10318],{"className":10294},[117],[114,10296,10298],{"className":10297},[121],[123,10299,10300],{"xmlns":125},[127,10301,10302,10315],{},[130,10303,10304,10306,10309,10311,10313],{},[138,10305,1956],{},[133,10307,10308],{},"−",[138,10310,5648],{},[133,10312,828],{},[138,10314,756],{},[157,10316,10317],{"encoding":159},"a-b \\in I",[114,10319,10321,10340,10359],{"className":10320,"ariaHidden":143},[164],[114,10322,10324,10328,10331,10334,10337],{"className":10323},[168],[114,10325],{"className":10326,"style":10327},[172],"height:0.6667em;vertical-align:-0.0833em;",[114,10329,1956],{"className":10330},[181,182],[114,10332],{"className":10333,"style":652},[191],[114,10335,10308],{"className":10336},[656],[114,10338],{"className":10339,"style":652},[191],[114,10341,10343,10347,10350,10353,10356],{"className":10342},[168],[114,10344],{"className":10345,"style":10346},[172],"height:0.7335em;vertical-align:-0.0391em;",[114,10348,5648],{"className":10349},[181,182],[114,10351],{"className":10352,"style":250},[191],[114,10354,828],{"className":10355},[254],[114,10357],{"className":10358,"style":250},[191],[114,10360,10362,10365],{"className":10361},[168],[114,10363],{"className":10364,"style":264},[172],[114,10366,756],{"className":10367,"style":771},[181,182],[114,10369,10371,10393],{"className":10370},[117],[114,10372,10374],{"className":10373},[121],[123,10375,10376],{"xmlns":125},[127,10377,10378,10390],{},[130,10379,10380,10382,10384,10386,10388],{},[138,10381,1956],{},[133,10383,10308],{},[138,10385,5648],{},[133,10387,828],{},[138,10389,9856],{},[157,10391,10392],{"encoding":159},"a-b \\in J",[114,10394,10396,10414,10432],{"className":10395,"ariaHidden":143},[164],[114,10397,10399,10402,10405,10408,10411],{"className":10398},[168],[114,10400],{"className":10401,"style":10327},[172],[114,10403,1956],{"className":10404},[181,182],[114,10406],{"className":10407,"style":652},[191],[114,10409,10308],{"className":10410},[656],[114,10412],{"className":10413,"style":652},[191],[114,10415,10417,10420,10423,10426,10429],{"className":10416},[168],[114,10418],{"className":10419,"style":10346},[172],[114,10421,5648],{"className":10422},[181,182],[114,10424],{"className":10425,"style":250},[191],[114,10427,828],{"className":10428},[254],[114,10430],{"className":10431,"style":250},[191],[114,10433,10435,10438],{"className":10434},[168],[114,10436],{"className":10437,"style":264},[172],[114,10439,9856],{"className":10440,"style":9881},[181,182],[114,10442,10444,10470],{"className":10443},[117],[114,10445,10447],{"className":10446},[121],[123,10448,10449],{"xmlns":125},[127,10450,10451,10467],{},[130,10452,10453,10455,10457,10459,10461,10463,10465],{},[138,10454,1956],{},[133,10456,10308],{},[138,10458,5648],{},[133,10460,828],{},[138,10462,756],{},[133,10464,9931],{},[138,10466,9856],{},[157,10468,10469],{"encoding":159},"a-b \\in I \\cap J",[114,10471,10473,10491,10509,10527],{"className":10472,"ariaHidden":143},[164],[114,10474,10476,10479,10482,10485,10488],{"className":10475},[168],[114,10477],{"className":10478,"style":10327},[172],[114,10480,1956],{"className":10481},[181,182],[114,10483],{"className":10484,"style":652},[191],[114,10486,10308],{"className":10487},[656],[114,10489],{"className":10490,"style":652},[191],[114,10492,10494,10497,10500,10503,10506],{"className":10493},[168],[114,10495],{"className":10496,"style":10346},[172],[114,10498,5648],{"className":10499},[181,182],[114,10501],{"className":10502,"style":250},[191],[114,10504,828],{"className":10505},[254],[114,10507],{"className":10508,"style":250},[191],[114,10510,10512,10515,10518,10521,10524],{"className":10511},[168],[114,10513],{"className":10514,"style":264},[172],[114,10516,756],{"className":10517,"style":771},[181,182],[114,10519],{"className":10520,"style":652},[191],[114,10522,9931],{"className":10523},[656],[114,10525],{"className":10526,"style":652},[191],[114,10528,10530,10533],{"className":10529},[168],[114,10531],{"className":10532,"style":264},[172],[114,10534,9856],{"className":10535,"style":9881},[181,182],[450,10537,10538,5343,10541,1595,10649],{},[51,10539,10540],{},"吸收律",[114,10542,10544,10574],{"className":10543},[117],[114,10545,10547],{"className":10546},[121],[123,10548,10549],{"xmlns":125},[127,10550,10551,10571],{},[130,10552,10553,10555,10557,10559,10561,10563,10565,10567,10569],{},[138,10554,835],{},[133,10556,828],{},[138,10558,756],{},[133,10560,9931],{},[138,10562,9856],{},[133,10564,144],{"separator":143},[138,10566,825],{},[133,10568,828],{},[138,10570,140],{},[157,10572,10573],{"encoding":159},"x \\in I \\cap J, r \\in R",[114,10575,10577,10595,10613,10640],{"className":10576,"ariaHidden":143},[164],[114,10578,10580,10583,10586,10589,10592],{"className":10579},[168],[114,10581],{"className":10582,"style":852},[172],[114,10584,835],{"className":10585},[181,182],[114,10587],{"className":10588,"style":250},[191],[114,10590,828],{"className":10591},[254],[114,10593],{"className":10594,"style":250},[191],[114,10596,10598,10601,10604,10607,10610],{"className":10597},[168],[114,10599],{"className":10600,"style":264},[172],[114,10602,756],{"className":10603,"style":771},[181,182],[114,10605],{"className":10606,"style":652},[191],[114,10608,9931],{"className":10609},[656],[114,10611],{"className":10612,"style":652},[191],[114,10614,10616,10619,10622,10625,10628,10631,10634,10637],{"className":10615},[168],[114,10617],{"className":10618,"style":872},[172],[114,10620,9856],{"className":10621,"style":9881},[181,182],[114,10623,144],{"className":10624},[187],[114,10626],{"className":10627,"style":192},[191],[114,10629,825],{"className":10630,"style":856},[181,182],[114,10632],{"className":10633,"style":250},[191],[114,10635,828],{"className":10636},[254],[114,10638],{"className":10639,"style":250},[191],[114,10641,10643,10646],{"className":10642},[168],[114,10644],{"className":10645,"style":264},[172],[114,10647,140],{"className":10648,"style":183},[181,182],[447,10650,10651,10850,11046,11145],{},[450,10652,10653,10184,10765,10820,10821,10849],{},[114,10654,10656,10689],{"className":10655},[117],[114,10657,10659],{"className":10658},[121],[123,10660,10661],{"xmlns":125},[127,10662,10663,10686],{},[130,10664,10665,10667,10669,10671,10673,10676,10678,10680,10682,10684],{},[138,10666,835],{},[133,10668,828],{},[138,10670,756],{},[6955,10672,7061],{},[133,10674,10675],{},"⟹",[6955,10677,7061],{},[138,10679,825],{},[138,10681,835],{},[133,10683,828],{},[138,10685,756],{},[157,10687,10688],{"encoding":159},"x \\in I \\implies rx \\in I",[114,10690,10692,10710,10735,10756],{"className":10691,"ariaHidden":143},[164],[114,10693,10695,10698,10701,10704,10707],{"className":10694},[168],[114,10696],{"className":10697,"style":852},[172],[114,10699,835],{"className":10700},[181,182],[114,10702],{"className":10703,"style":250},[191],[114,10705,828],{"className":10706},[254],[114,10708],{"className":10709,"style":250},[191],[114,10711,10713,10717,10720,10723,10726,10729,10732],{"className":10712},[168],[114,10714],{"className":10715,"style":10716},[172],"height:0.7073em;vertical-align:-0.024em;",[114,10718,756],{"className":10719,"style":771},[181,182],[114,10721],{"className":10722,"style":250},[191],[114,10724],{"className":10725,"style":250},[191],[114,10727,10675],{"className":10728},[254],[114,10730],{"className":10731,"style":250},[191],[114,10733],{"className":10734,"style":250},[191],[114,10736,10738,10741,10744,10747,10750,10753],{"className":10737},[168],[114,10739],{"className":10740,"style":852},[172],[114,10742,825],{"className":10743,"style":856},[181,182],[114,10745,835],{"className":10746},[181,182],[114,10748],{"className":10749,"style":250},[191],[114,10751,828],{"className":10752},[254],[114,10754],{"className":10755,"style":250},[191],[114,10757,10759,10762],{"className":10758},[168],[114,10760],{"className":10761,"style":264},[172],[114,10763,756],{"className":10764,"style":771},[181,182],[114,10766,10768,10787],{"className":10767},[117],[114,10769,10771],{"className":10770},[121],[123,10772,10773],{"xmlns":125},[127,10774,10775,10785],{},[130,10776,10777,10779,10781,10783],{},[138,10778,835],{},[138,10780,825],{},[133,10782,828],{},[138,10784,756],{},[157,10786,1135],{"encoding":159},[114,10788,10790,10811],{"className":10789,"ariaHidden":143},[164],[114,10791,10793,10796,10799,10802,10805,10808],{"className":10792},[168],[114,10794],{"className":10795,"style":852},[172],[114,10797,835],{"className":10798},[181,182],[114,10800,825],{"className":10801,"style":856},[181,182],[114,10803],{"className":10804,"style":250},[191],[114,10806,828],{"className":10807},[254],[114,10809],{"className":10810,"style":250},[191],[114,10812,10814,10817],{"className":10813},[168],[114,10815],{"className":10816,"style":264},[172],[114,10818,756],{"className":10819,"style":771},[181,182],"（因 ",[114,10822,10824,10837],{"className":10823},[117],[114,10825,10827],{"className":10826},[121],[123,10828,10829],{"xmlns":125},[127,10830,10831,10835],{},[130,10832,10833],{},[138,10834,756],{},[157,10836,756],{"encoding":159},[114,10838,10840],{"className":10839,"ariaHidden":143},[164],[114,10841,10843,10846],{"className":10842},[168],[114,10844],{"className":10845,"style":264},[172],[114,10847,756],{"className":10848,"style":771},[181,182]," 是理想）。",[450,10851,10852,10184,10962,10820,11018,10849],{},[114,10853,10855,10887],{"className":10854},[117],[114,10856,10858],{"className":10857},[121],[123,10859,10860],{"xmlns":125},[127,10861,10862,10884],{},[130,10863,10864,10866,10868,10870,10872,10874,10876,10878,10880,10882],{},[138,10865,835],{},[133,10867,828],{},[138,10869,9856],{},[6955,10871,7061],{},[133,10873,10675],{},[6955,10875,7061],{},[138,10877,825],{},[138,10879,835],{},[133,10881,828],{},[138,10883,9856],{},[157,10885,10886],{"encoding":159},"x \\in J \\implies rx \\in J",[114,10888,10890,10908,10932,10953],{"className":10889,"ariaHidden":143},[164],[114,10891,10893,10896,10899,10902,10905],{"className":10892},[168],[114,10894],{"className":10895,"style":852},[172],[114,10897,835],{"className":10898},[181,182],[114,10900],{"className":10901,"style":250},[191],[114,10903,828],{"className":10904},[254],[114,10906],{"className":10907,"style":250},[191],[114,10909,10911,10914,10917,10920,10923,10926,10929],{"className":10910},[168],[114,10912],{"className":10913,"style":10716},[172],[114,10915,9856],{"className":10916,"style":9881},[181,182],[114,10918],{"className":10919,"style":250},[191],[114,10921],{"className":10922,"style":250},[191],[114,10924,10675],{"className":10925},[254],[114,10927],{"className":10928,"style":250},[191],[114,10930],{"className":10931,"style":250},[191],[114,10933,10935,10938,10941,10944,10947,10950],{"className":10934},[168],[114,10936],{"className":10937,"style":852},[172],[114,10939,825],{"className":10940,"style":856},[181,182],[114,10942,835],{"className":10943},[181,182],[114,10945],{"className":10946,"style":250},[191],[114,10948,828],{"className":10949},[254],[114,10951],{"className":10952,"style":250},[191],[114,10954,10956,10959],{"className":10955},[168],[114,10957],{"className":10958,"style":264},[172],[114,10960,9856],{"className":10961,"style":9881},[181,182],[114,10963,10965,10985],{"className":10964},[117],[114,10966,10968],{"className":10967},[121],[123,10969,10970],{"xmlns":125},[127,10971,10972,10982],{},[130,10973,10974,10976,10978,10980],{},[138,10975,835],{},[138,10977,825],{},[133,10979,828],{},[138,10981,9856],{},[157,10983,10984],{"encoding":159},"xr \\in J",[114,10986,10988,11009],{"className":10987,"ariaHidden":143},[164],[114,10989,10991,10994,10997,11000,11003,11006],{"className":10990},[168],[114,10992],{"className":10993,"style":852},[172],[114,10995,835],{"className":10996},[181,182],[114,10998,825],{"className":10999,"style":856},[181,182],[114,11001],{"className":11002,"style":250},[191],[114,11004,828],{"className":11005},[254],[114,11007],{"className":11008,"style":250},[191],[114,11010,11012,11015],{"className":11011},[168],[114,11013],{"className":11014,"style":264},[172],[114,11016,9856],{"className":11017,"style":9881},[181,182],[114,11019,11021,11034],{"className":11020},[117],[114,11022,11024],{"className":11023},[121],[123,11025,11026],{"xmlns":125},[127,11027,11028,11032],{},[130,11029,11030],{},[138,11031,9856],{},[157,11033,9856],{"encoding":159},[114,11035,11037],{"className":11036,"ariaHidden":143},[164],[114,11038,11040,11043],{"className":11039},[168],[114,11041],{"className":11042,"style":264},[172],[114,11044,9856],{"className":11045,"style":9881},[181,182],[450,11047,6225,11048,445],{},[114,11049,11051,11081],{"className":11050},[117],[114,11052,11054],{"className":11053},[121],[123,11055,11056],{"xmlns":125},[127,11057,11058,11078],{},[130,11059,11060,11062,11064,11066,11068,11070,11072,11074,11076],{},[138,11061,825],{},[138,11063,835],{},[133,11065,144],{"separator":143},[138,11067,835],{},[138,11069,825],{},[133,11071,828],{},[138,11073,756],{},[133,11075,9931],{},[138,11077,9856],{},[157,11079,11080],{"encoding":159},"rx, xr \\in I \\cap J",[114,11082,11084,11118,11136],{"className":11083,"ariaHidden":143},[164],[114,11085,11087,11091,11094,11097,11100,11103,11106,11109,11112,11115],{"className":11086},[168],[114,11088],{"className":11089,"style":11090},[172],"height:0.7335em;vertical-align:-0.1944em;",[114,11092,825],{"className":11093,"style":856},[181,182],[114,11095,835],{"className":11096},[181,182],[114,11098,144],{"className":11099},[187],[114,11101],{"className":11102,"style":192},[191],[114,11104,835],{"className":11105},[181,182],[114,11107,825],{"className":11108,"style":856},[181,182],[114,11110],{"className":11111,"style":250},[191],[114,11113,828],{"className":11114},[254],[114,11116],{"className":11117,"style":250},[191],[114,11119,11121,11124,11127,11130,11133],{"className":11120},[168],[114,11122],{"className":11123,"style":264},[172],[114,11125,756],{"className":11126,"style":771},[181,182],[114,11128],{"className":11129,"style":652},[191],[114,11131,9931],{"className":11132},[656],[114,11134],{"className":11135,"style":652},[191],[114,11137,11139,11142],{"className":11138},[168],[114,11140],{"className":11141,"style":264},[172],[114,11143,9856],{"className":11144,"style":9881},[181,182],[450,11146,11147,536,11150,11200],{},[51,11148,11149],{},"结论",[114,11151,11153,11170],{"className":11152},[117],[114,11154,11156],{"className":11155},[121],[123,11157,11158],{"xmlns":125},[127,11159,11160,11168],{},[130,11161,11162,11164,11166],{},[138,11163,756],{},[133,11165,9931],{},[138,11167,9856],{},[157,11169,9936],{"encoding":159},[114,11171,11173,11191],{"className":11172,"ariaHidden":143},[164],[114,11174,11176,11179,11182,11185,11188],{"className":11175},[168],[114,11177],{"className":11178,"style":264},[172],[114,11180,756],{"className":11181,"style":771},[181,182],[114,11183],{"className":11184,"style":652},[191],[114,11186,9931],{"className":11187},[656],[114,11189],{"className":11190,"style":652},[191],[114,11192,11194,11197],{"className":11193},[168],[114,11195],{"className":11196,"style":264},[172],[114,11198,9856],{"className":11199,"style":9881},[181,182]," 是理想。",[106,11202,11204,11205],{"id":11203},"_52-和-iji-jij","5.2 和 ",[114,11206,11208,11226],{"className":11207},[117],[114,11209,11211],{"className":11210},[121],[123,11212,11213],{"xmlns":125},[127,11214,11215,11223],{},[130,11216,11217,11219,11221],{},[138,11218,756],{},[133,11220,147],{},[138,11222,9856],{},[157,11224,11225],{"encoding":159},"I + J",[114,11227,11229,11248],{"className":11228,"ariaHidden":143},[164],[114,11230,11232,11236,11239,11242,11245],{"className":11231},[168],[114,11233],{"className":11234,"style":11235},[172],"height:0.7667em;vertical-align:-0.0833em;",[114,11237,756],{"className":11238,"style":771},[181,182],[114,11240],{"className":11241,"style":652},[191],[114,11243,147],{"className":11244},[656],[114,11246],{"className":11247,"style":652},[191],[114,11249,11251,11254],{"className":11250},[168],[114,11252],{"className":11253,"style":264},[172],[114,11255,9856],{"className":11256,"style":9881},[181,182],[19,11258,11259,536,11262,1595,11446,536,11448,10022,11498,536],{},[51,11260,11261],{},"定义",[114,11263,11265,11311],{"className":11264},[117],[114,11266,11268],{"className":11267},[121],[123,11269,11270],{"xmlns":125},[127,11271,11272,11308],{},[130,11273,11274,11276,11278,11280,11282,11284,11286,11288,11290,11292,11294,11296,11298,11300,11302,11304,11306],{},[138,11275,756],{},[133,11277,147],{},[138,11279,9856],{},[133,11281,1403],{},[133,11283,626],{"stretchy":135},[138,11285,1956],{},[133,11287,147],{},[138,11289,5648],{},[133,11291,9740],{},[138,11293,1956],{},[133,11295,828],{},[138,11297,756],{},[133,11299,144],{"separator":143},[138,11301,5648],{},[133,11303,828],{},[138,11305,9856],{},[133,11307,632],{"stretchy":135},[157,11309,11310],{"encoding":159},"I + J = \\{ a + b \\mid a \\in I, b \\in J \\}",[114,11312,11314,11332,11350,11371,11389,11407,11434],{"className":11313,"ariaHidden":143},[164],[114,11315,11317,11320,11323,11326,11329],{"className":11316},[168],[114,11318],{"className":11319,"style":11235},[172],[114,11321,756],{"className":11322,"style":771},[181,182],[114,11324],{"className":11325,"style":652},[191],[114,11327,147],{"className":11328},[656],[114,11330],{"className":11331,"style":652},[191],[114,11333,11335,11338,11341,11344,11347],{"className":11334},[168],[114,11336],{"className":11337,"style":264},[172],[114,11339,9856],{"className":11340,"style":9881},[181,182],[114,11342],{"className":11343,"style":250},[191],[114,11345,1403],{"className":11346},[254],[114,11348],{"className":11349,"style":250},[191],[114,11351,11353,11356,11359,11362,11365,11368],{"className":11352},[168],[114,11354],{"className":11355,"style":173},[172],[114,11357,626],{"className":11358},[177],[114,11360,1956],{"className":11361},[181,182],[114,11363],{"className":11364,"style":652},[191],[114,11366,147],{"className":11367},[656],[114,11369],{"className":11370,"style":652},[191],[114,11372,11374,11377,11380,11383,11386],{"className":11373},[168],[114,11375],{"className":11376,"style":173},[172],[114,11378,5648],{"className":11379},[181,182],[114,11381],{"className":11382,"style":250},[191],[114,11384,9740],{"className":11385},[254],[114,11387],{"className":11388,"style":250},[191],[114,11390,11392,11395,11398,11401,11404],{"className":11391},[168],[114,11393],{"className":11394,"style":852},[172],[114,11396,1956],{"className":11397},[181,182],[114,11399],{"className":11400,"style":250},[191],[114,11402,828],{"className":11403},[254],[114,11405],{"className":11406,"style":250},[191],[114,11408,11410,11413,11416,11419,11422,11425,11428,11431],{"className":11409},[168],[114,11411],{"className":11412,"style":1810},[172],[114,11414,756],{"className":11415,"style":771},[181,182],[114,11417,144],{"className":11418},[187],[114,11420],{"className":11421,"style":192},[191],[114,11423,5648],{"className":11424},[181,182],[114,11426],{"className":11427,"style":250},[191],[114,11429,828],{"className":11430},[254],[114,11432],{"className":11433,"style":250},[191],[114,11435,11437,11440,11443],{"className":11436},[168],[114,11438],{"className":11439,"style":173},[172],[114,11441,9856],{"className":11442,"style":9881},[181,182],[114,11444,632],{"className":11445},[208],[51,11447,9971],{},[114,11449,11451,11468],{"className":11450},[117],[114,11452,11454],{"className":11453},[121],[123,11455,11456],{"xmlns":125},[127,11457,11458,11466],{},[130,11459,11460,11462,11464],{},[138,11461,756],{},[133,11463,147],{},[138,11465,9856],{},[157,11467,11225],{"encoding":159},[114,11469,11471,11489],{"className":11470,"ariaHidden":143},[164],[114,11472,11474,11477,11480,11483,11486],{"className":11473},[168],[114,11475],{"className":11476,"style":11235},[172],[114,11478,756],{"className":11479,"style":771},[181,182],[114,11481],{"className":11482,"style":652},[191],[114,11484,147],{"className":11485},[656],[114,11487],{"className":11488,"style":652},[191],[114,11490,11492,11495],{"className":11491},[168],[114,11493],{"className":11494,"style":264},[172],[114,11496,9856],{"className":11497,"style":9881},[181,182],[51,11499,10025],{},[803,11501,11502,12580],{},[450,11503,11504,5343,11506,1595,11847,12170,12171,12212,12213,12527,12528,12579],{},[51,11505,10032],{},[114,11507,11509,11568],{"className":11508},[117],[114,11510,11512],{"className":11511},[121],[123,11513,11514],{"xmlns":125},[127,11515,11516,11565],{},[130,11517,11518,11520,11522,11528,11530,11536,11538,11541,11543,11549,11551,11557,11559,11561,11563],{},[138,11519,835],{},[133,11521,1403],{},[1952,11523,11524,11526],{},[138,11525,1956],{},[550,11527,4373],{},[133,11529,147],{},[1952,11531,11532,11534],{},[138,11533,5648],{},[550,11535,4373],{},[133,11537,144],{"separator":143},[138,11539,11540],{},"y",[133,11542,1403],{},[1952,11544,11545,11547],{},[138,11546,1956],{},[550,11548,552],{},[133,11550,147],{},[1952,11552,11553,11555],{},[138,11554,5648],{},[550,11556,552],{},[133,11558,828],{},[138,11560,756],{},[133,11562,147],{},[138,11564,9856],{},[157,11566,11567],{"encoding":159},"x = a_1 + b_1, y = a_2 + b_2 \\in I+J",[114,11569,11571,11589,11645,11709,11764,11820,11838],{"className":11570,"ariaHidden":143},[164],[114,11572,11574,11577,11580,11583,11586],{"className":11573},[168],[114,11575],{"className":11576,"style":1420},[172],[114,11578,835],{"className":11579},[181,182],[114,11581],{"className":11582,"style":250},[191],[114,11584,1403],{"className":11585},[254],[114,11587],{"className":11588,"style":250},[191],[114,11590,11592,11596,11636,11639,11642],{"className":11591},[168],[114,11593],{"className":11594,"style":11595},[172],"height:0.7333em;vertical-align:-0.15em;",[114,11597,11599,11602],{"className":11598},[181],[114,11600,1956],{"className":11601},[181,182],[114,11603,11605],{"className":11604},[2010],[114,11606,11608,11628],{"className":11607},[2014,2015],[114,11609,11611,11625],{"className":11610},[2019],[114,11612,11614],{"className":11613,"style":2829},[2023],[114,11615,11616,11619],{"style":2027},[114,11617],{"className":11618,"style":2032},[2031],[114,11620,11622],{"className":11621},[2036,2037,2038,2039],[114,11623,4373],{"className":11624},[181,2039],[114,11626,2052],{"className":11627},[2051],[114,11629,11631],{"className":11630},[2019],[114,11632,11634],{"className":11633,"style":2850},[2023],[114,11635],{},[114,11637],{"className":11638,"style":652},[191],[114,11640,147],{"className":11641},[656],[114,11643],{"className":11644,"style":652},[191],[114,11646,11648,11651,11691,11694,11697,11700,11703,11706],{"className":11647},[168],[114,11649],{"className":11650,"style":1810},[172],[114,11652,11654,11657],{"className":11653},[181],[114,11655,5648],{"className":11656},[181,182],[114,11658,11660],{"className":11659},[2010],[114,11661,11663,11683],{"className":11662},[2014,2015],[114,11664,11666,11680],{"className":11665},[2019],[114,11667,11669],{"className":11668,"style":2829},[2023],[114,11670,11671,11674],{"style":2027},[114,11672],{"className":11673,"style":2032},[2031],[114,11675,11677],{"className":11676},[2036,2037,2038,2039],[114,11678,4373],{"className":11679},[181,2039],[114,11681,2052],{"className":11682},[2051],[114,11684,11686],{"className":11685},[2019],[114,11687,11689],{"className":11688,"style":2850},[2023],[114,11690],{},[114,11692,144],{"className":11693},[187],[114,11695],{"className":11696,"style":192},[191],[114,11698,11540],{"className":11699,"style":2453},[181,182],[114,11701],{"className":11702,"style":250},[191],[114,11704,1403],{"className":11705},[254],[114,11707],{"className":11708,"style":250},[191],[114,11710,11712,11715,11755,11758,11761],{"className":11711},[168],[114,11713],{"className":11714,"style":11595},[172],[114,11716,11718,11721],{"className":11717},[181],[114,11719,1956],{"className":11720},[181,182],[114,11722,11724],{"className":11723},[2010],[114,11725,11727,11747],{"className":11726},[2014,2015],[114,11728,11730,11744],{"className":11729},[2019],[114,11731,11733],{"className":11732,"style":2829},[2023],[114,11734,11735,11738],{"style":2027},[114,11736],{"className":11737,"style":2032},[2031],[114,11739,11741],{"className":11740},[2036,2037,2038,2039],[114,11742,552],{"className":11743},[181,2039],[114,11745,2052],{"className":11746},[2051],[114,11748,11750],{"className":11749},[2019],[114,11751,11753],{"className":11752,"style":2850},[2023],[114,11754],{},[114,11756],{"className":11757,"style":652},[191],[114,11759,147],{"className":11760},[656],[114,11762],{"className":11763,"style":652},[191],[114,11765,11767,11771,11811,11814,11817],{"className":11766},[168],[114,11768],{"className":11769,"style":11770},[172],"height:0.8444em;vertical-align:-0.15em;",[114,11772,11774,11777],{"className":11773},[181],[114,11775,5648],{"className":11776},[181,182],[114,11778,11780],{"className":11779},[2010],[114,11781,11783,11803],{"className":11782},[2014,2015],[114,11784,11786,11800],{"className":11785},[2019],[114,11787,11789],{"className":11788,"style":2829},[2023],[114,11790,11791,11794],{"style":2027},[114,11792],{"className":11793,"style":2032},[2031],[114,11795,11797],{"className":11796},[2036,2037,2038,2039],[114,11798,552],{"className":11799},[181,2039],[114,11801,2052],{"className":11802},[2051],[114,11804,11806],{"className":11805},[2019],[114,11807,11809],{"className":11808,"style":2850},[2023],[114,11810],{},[114,11812],{"className":11813,"style":250},[191],[114,11815,828],{"className":11816},[254],[114,11818],{"className":11819,"style":250},[191],[114,11821,11823,11826,11829,11832,11835],{"className":11822},[168],[114,11824],{"className":11825,"style":11235},[172],[114,11827,756],{"className":11828,"style":771},[181,182],[114,11830],{"className":11831,"style":652},[191],[114,11833,147],{"className":11834},[656],[114,11836],{"className":11837,"style":652},[191],[114,11839,11841,11844],{"className":11840},[168],[114,11842],{"className":11843,"style":264},[172],[114,11845,9856],{"className":11846,"style":9881},[181,182],[114,11848,11850,11908],{"className":11849},[117],[114,11851,11853],{"className":11852},[121],[123,11854,11855],{"xmlns":125},[127,11856,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- y = (a_1 - a_2) + (b_1 - b_2)",[114,11909,11911,11929,11947,12005,12063,12121],{"className":11910,"ariaHidden":143},[164],[114,11912,11914,11917,11920,11923,11926],{"className":11913},[168],[114,11915],{"className":11916,"style":10327},[172],[114,11918,835],{"className":11919},[181,182],[114,11921],{"className":11922,"style":652},[191],[114,11924,10308],{"className":11925},[656],[114,11927],{"className":11928,"style":652},[191],[114,11930,11932,11935,11938,11941,11944],{"className":11931},[168],[114,11933],{"className":11934,"style":2492},[172],[114,11936,11540],{"className":11937,"style":2453},[181,182],[114,11939],{"className":11940,"style":250},[191],[114,11942,1403],{"className":11943},[254],[114,11945],{"className":11946,"style":250},[191],[114,11948,11950,11953,11956,11996,11999,12002],{"className":11949},[168],[114,11951],{"className":11952,"style":173},[172],[114,11954,136],{"className":11955},[177],[114,11957,11959,11962],{"className":11958},[181],[114,11960,1956],{"className":11961},[181,182],[114,11963,11965],{"className":11964},[2010],[114,11966,11968,11988],{"className":11967},[2014,2015],[114,11969,11971,11985],{"className":11970},[2019],[114,11972,11974],{"className":11973,"style":2829},[2023],[114,11975,11976,11979],{"style":2027},[114,11977],{"className":11978,"style":2032},[2031],[114,11980,11982],{"className":11981},[2036,2037,2038,2039],[114,11983,4373],{"className":11984},[181,2039],[114,11986,2052],{"className":11987},[2051],[114,11989,11991],{"className":11990},[2019],[114,11992,11994],{"className":11993,"style":2850},[2023],[114,11995],{},[114,11997],{"className":11998,"style":652},[191],[114,12000,10308],{"className":12001},[656],[114,12003],{"className":12004,"style":652},[191],[114,12006,12008,12011,12051,12054,12057,12060],{"className":12007},[168],[114,12009],{"className":12010,"style":173},[172],[114,12012,12014,12017],{"className":12013},[181],[114,12015,1956],{"className":12016},[181,182],[114,12018,12020],{"className":12019},[2010],[114,12021,12023,12043],{"className":12022},[2014,2015],[114,12024,12026,12040],{"className":12025},[2019],[114,12027,12029],{"className":12028,"style":2829},[2023],[114,12030,12031,12034],{"style":2027},[114,12032],{"className":12033,"style":2032},[2031],[114,12035,12037],{"className":12036},[2036,2037,2038,2039],[114,12038,552],{"className":12039},[181,2039],[114,12041,2052],{"className":12042},[2051],[114,12044,12046],{"className":12045},[2019],[114,12047,12049],{"className":12048,"style":2850},[2023],[114,12050],{},[114,12052,155],{"className":12053},[208],[114,12055],{"className":12056,"style":652},[191],[114,12058,147],{"className":12059},[656],[114,12061],{"className":12062,"style":652},[191],[114,12064,12066,12069,12072,12112,12115,12118],{"className":12065},[168],[114,12067],{"className":12068,"style":173},[172],[114,12070,136],{"className":12071},[177],[114,12073,12075,12078],{"className":12074},[181],[114,12076,5648],{"className":12077},[181,182],[114,12079,12081],{"className":12080},[2010],[114,12082,12084,12104],{"className":12083},[2014,2015],[114,12085,12087,12101],{"className":12086},[2019],[114,12088,12090],{"className":12089,"style":2829},[2023],[114,12091,12092,12095],{"style":2027},[114,12093],{"className":12094,"style":2032},[2031],[114,12096,12098],{"className":12097},[2036,2037,2038,2039],[114,12099,4373],{"className":12100},[181,2039],[114,12102,2052],{"className":12103},[2051],[114,12105,12107],{"className":12106},[2019],[114,12108,12110],{"className":12109,"style":2850},[2023],[114,12111],{},[114,12113],{"className":12114,"style":652},[191],[114,12116,10308],{"className":12117},[656],[114,12119],{"className":12120,"style":652},[191],[114,12122,12124,12127,12167],{"className":12123},[168],[114,12125],{"className":12126,"style":173},[172],[114,12128,12130,12133],{"className":12129},[181],[114,12131,5648],{"className":12132},[181,182],[114,12134,12136],{"className":12135},[2010],[114,12137,12139,12159],{"className":12138},[2014,2015],[114,12140,12142,12156],{"className":12141},[2019],[114,12143,12145],{"className":12144,"style":2829},[2023],[114,12146,12147,12150],{"style":2027},[114,12148],{"className":12149,"style":2032},[2031],[114,12151,12153],{"className":12152},[2036,2037,2038,2039],[114,12154,552],{"className":12155},[181,2039],[114,12157,2052],{"className":12158},[2051],[114,12160,12162],{"className":12161},[2019],[114,12163,12165],{"className":12164,"style":2850},[2023],[114,12166],{},[114,12168,155],{"className":12169},[208],"。由于 ",[114,12172,12174,12191],{"className":12173},[117],[114,12175,12177],{"className":12176},[121],[123,12178,12179],{"xmlns":125},[127,12180,12181,12189],{},[130,12182,12183,12185,12187],{},[138,12184,756],{},[133,12186,144],{"separator":143},[138,12188,9856],{},[157,12190,9859],{"encoding":159},[114,12192,12194],{"className":12193,"ariaHidden":143},[164],[114,12195,12197,12200,12203,12206,12209],{"className":12196},[168],[114,12198],{"className":12199,"style":872},[172],[114,12201,756],{"className":12202,"style":771},[181,182],[114,12204,144],{"className":12205},[187],[114,12207],{"className":12208,"style":192},[191],[114,12210,9856],{"className":12211,"style":9881},[181,182]," 是子群，",[114,12214,12216,12274],{"className":12215},[117],[114,12217,12219],{"className":12218},[121],[123,12220,12221],{"xmlns":125},[127,12222,12223,12271],{},[130,12224,12225,12227,12233,12235,12241,12243,12245,12247,12249,12251,12257,12259,12265,12267,12269],{},[133,12226,136],{"stretchy":135},[1952,12228,12229,12231],{},[138,12230,1956],{},[550,12232,4373],{},[133,12234,10308],{},[1952,12236,12237,12239],{},[138,12238,1956],{},[550,12240,552],{},[133,12242,155],{"stretchy":135},[133,12244,828],{},[138,12246,756],{},[133,12248,144],{"separator":143},[133,12250,136],{"stretchy":135},[1952,12252,12253,12255],{},[138,12254,5648],{},[550,12256,4373],{},[133,12258,10308],{},[1952,12260,12261,12263],{},[138,12262,5648],{},[550,12264,552],{},[133,12266,155],{"stretchy":135},[133,12268,828],{},[138,12270,9856],{},[157,12272,12273],{"encoding":159},"(a_1 - a_2) \\in I, (b_1 - b_2) \\in J",[114,12275,12277,12335,12393,12460,12518],{"className":12276,"ariaHidden":143},[164],[114,12278,12280,12283,12286,12326,12329,12332],{"className":12279},[168],[114,12281],{"className":12282,"style":173},[172],[114,12284,136],{"className":12285},[177],[114,12287,12289,12292],{"className":12288},[181],[114,12290,1956],{"className":12291},[181,182],[114,12293,12295],{"className":12294},[2010],[114,12296,12298,12318],{"className":12297},[2014,2015],[114,12299,12301,12315],{"className":12300},[2019],[114,12302,12304],{"className":12303,"style":2829},[2023],[114,12305,12306,12309],{"style":2027},[114,12307],{"className":12308,"style":2032},[2031],[114,12310,12312],{"className":12311},[2036,2037,2038,2039],[114,12313,4373],{"className":12314},[181,2039],[114,12316,2052],{"className":12317},[2051],[114,12319,12321],{"className":12320},[2019],[114,12322,12324],{"className":12323,"style":2850},[2023],[114,12325],{},[114,12327],{"className":12328,"style":652},[191],[114,12330,10308],{"className":12331},[656],[114,12333],{"className":12334,"style":652},[191],[114,12336,12338,12341,12381,12384,12387,12390],{"className":12337},[168],[114,12339],{"className":12340,"style":173},[172],[114,12342,12344,12347],{"className":12343},[181],[114,12345,1956],{"className":12346},[181,182],[114,12348,12350],{"className":12349},[2010],[114,12351,12353,12373],{"className":12352},[2014,2015],[114,12354,12356,12370],{"className":12355},[2019],[114,12357,12359],{"className":12358,"style":2829},[2023],[114,12360,12361,12364],{"style":2027},[114,12362],{"className":12363,"style":2032},[2031],[114,12365,12367],{"className":12366},[2036,2037,2038,2039],[114,12368,552],{"className":12369},[181,2039],[114,12371,2052],{"className":12372},[2051],[114,12374,12376],{"className":12375},[2019],[114,12377,12379],{"className":12378,"style":2850},[2023],[114,12380],{},[114,12382,155],{"className":12383},[208],[114,12385],{"className":12386,"style":250},[191],[114,12388,828],{"className":12389},[254],[114,12391],{"className":12392,"style":250},[191],[114,12394,12396,12399,12402,12405,12408,12411,12451,12454,12457],{"className":12395},[168],[114,12397],{"className":12398,"style":173},[172],[114,12400,756],{"className":12401,"style":771},[181,182],[114,12403,144],{"className":12404},[187],[114,12406],{"className":12407,"style":192},[191],[114,12409,136],{"className":12410},[177],[114,12412,12414,12417],{"className":12413},[181],[114,12415,5648],{"className":12416},[181,182],[114,12418,12420],{"className":12419},[2010],[114,12421,12423,12443],{"className":12422},[2014,2015],[114,12424,12426,12440],{"className":12425},[2019],[114,12427,12429],{"className":12428,"style":2829},[2023],[114,12430,12431,12434],{"style":2027},[114,12432],{"className":12433,"style":2032},[2031],[114,12435,12437],{"className":12436},[2036,2037,2038,2039],[114,12438,4373],{"className":12439},[181,2039],[114,12441,2052],{"className":12442},[2051],[114,12444,12446],{"className":12445},[2019],[114,12447,12449],{"className":12448,"style":2850},[2023],[114,12450],{},[114,12452],{"className":12453,"style":652},[191],[114,12455,10308],{"className":12456},[656],[114,12458],{"className":12459,"style":652},[191],[114,12461,12463,12466,12506,12509,12512,12515],{"className":12462},[168],[114,12464],{"className":12465,"style":173},[172],[114,12467,12469,12472],{"className":12468},[181],[114,12470,5648],{"className":12471},[181,182],[114,12473,12475],{"className":12474},[2010],[114,12476,12478,12498],{"className":12477},[2014,2015],[114,12479,12481,12495],{"className":12480},[2019],[114,12482,12484],{"className":12483,"style":2829},[2023],[114,12485,12486,12489],{"style":2027},[114,12487],{"className":12488,"style":2032},[2031],[114,12490,12492],{"className":12491},[2036,2037,2038,2039],[114,12493,552],{"className":12494},[181,2039],[114,12496,2052],{"className":12497},[2051],[114,12499,12501],{"className":12500},[2019],[114,12502,12504],{"className":12503,"style":2850},[2023],[114,12505],{},[114,12507,155],{"className":12508},[208],[114,12510],{"className":12511,"style":250},[191],[114,12513,828],{"className":12514},[254],[114,12516],{"className":12517,"style":250},[191],[114,12519,12521,12524],{"className":12520},[168],[114,12522],{"className":12523,"style":264},[172],[114,12525,9856],{"className":12526,"style":9881},[181,182],"。故差仍在 ",[114,12529,12531,12549],{"className":12530},[117],[114,12532,12534],{"className":12533},[121],[123,12535,12536],{"xmlns":125},[127,12537,12538,12546],{},[130,12539,12540,12542,12544],{},[138,12541,756],{},[133,12543,147],{},[138,12545,9856],{},[157,12547,12548],{"encoding":159},"I+J",[114,12550,12552,12570],{"className":12551,"ariaHidden":143},[164],[114,12553,12555,12558,12561,12564,12567],{"className":12554},[168],[114,12556],{"className":12557,"style":11235},[172],[114,12559,756],{"className":12560,"style":771},[181,182],[114,12562],{"className":12563,"style":652},[191],[114,12565,147],{"className":12566},[656],[114,12568],{"className":12569,"style":652},[191],[114,12571,12573,12576],{"className":12572},[168],[114,12574],{"className":12575,"style":264},[172],[114,12577,9856],{"className":12578,"style":9881},[181,182]," 中。",[450,12581,12582,5343,12584,12701,12702,1595,12753,12900,12901,12929,12930,10249,13011,12929,13039,13119,13120,13198,13199],{},[51,12583,10540],{},[114,12585,12587,12617],{"className":12586},[117],[114,12588,12590],{"className":12589},[121],[123,12591,12592],{"xmlns":125},[127,12593,12594,12614],{},[130,12595,12596,12598,12600,12602,12604,12606,12608,12610,12612],{},[138,12597,835],{},[133,12599,1403],{},[138,12601,1956],{},[133,12603,147],{},[138,12605,5648],{},[133,12607,828],{},[138,12609,756],{},[133,12611,147],{},[138,12613,9856],{},[157,12615,12616],{"encoding":159},"x = a + b \\in I+J",[114,12618,12620,12638,12656,12674,12692],{"className":12619,"ariaHidden":143},[164],[114,12621,12623,12626,12629,12632,12635],{"className":12622},[168],[114,12624],{"className":12625,"style":1420},[172],[114,12627,835],{"className":12628},[181,182],[114,12630],{"className":12631,"style":250},[191],[114,12633,1403],{"className":12634},[254],[114,12636],{"className":12637,"style":250},[191],[114,12639,12641,12644,12647,12650,12653],{"className":12640},[168],[114,12642],{"className":12643,"style":10327},[172],[114,12645,1956],{"className":12646},[181,182],[114,12648],{"className":12649,"style":652},[191],[114,12651,147],{"className":12652},[656],[114,12654],{"className":12655,"style":652},[191],[114,12657,12659,12662,12665,12668,12671],{"className":12658},[168],[114,12660],{"className":12661,"style":10346},[172],[114,12663,5648],{"className":12664},[181,182],[114,12666],{"className":12667,"style":250},[191],[114,12669,828],{"className":12670},[254],[114,12672],{"className":12673,"style":250},[191],[114,12675,12677,12680,12683,12686,12689],{"className":12676},[168],[114,12678],{"className":12679,"style":11235},[172],[114,12681,756],{"className":12682,"style":771},[181,182],[114,12684],{"className":12685,"style":652},[191],[114,12687,147],{"className":12688},[656],[114,12690],{"className":12691,"style":652},[191],[114,12693,12695,12698],{"className":12694},[168],[114,12696],{"className":12697,"style":264},[172],[114,12699,9856],{"className":12700,"style":9881},[181,182],"，",[114,12703,12705,12723],{"className":12704},[117],[114,12706,12708],{"className":12707},[121],[123,12709,12710],{"xmlns":125},[127,12711,12712,12720],{},[130,12713,12714,12716,12718],{},[138,12715,825],{},[133,12717,828],{},[138,12719,140],{},[157,12721,12722],{"encoding":159},"r \\in R",[114,12724,12726,12744],{"className":12725,"ariaHidden":143},[164],[114,12727,12729,12732,12735,12738,12741],{"className":12728},[168],[114,12730],{"className":12731,"style":852},[172],[114,12733,825],{"className":12734,"style":856},[181,182],[114,12736],{"className":12737,"style":250},[191],[114,12739,828],{"className":12740},[254],[114,12742],{"className":12743,"style":250},[191],[114,12745,12747,12750],{"className":12746},[168],[114,12748],{"className":12749,"style":264},[172],[114,12751,140],{"className":12752,"style":183},[181,182],[114,12754,12756,12798],{"className":12755},[117],[114,12757,12759],{"className":12758},[121],[123,12760,12761],{"xmlns":125},[127,12762,12763,12795],{},[130,12764,12765,12767,12769,12771,12773,12775,12777,12779,12781,12783,12785,12787,12789,12791,12793],{},[138,12766,825],{},[138,12768,835],{},[133,12770,1403],{},[138,12772,825],{},[133,12774,136],{"stretchy":135},[138,12776,1956],{},[133,12778,147],{},[138,12780,5648],{},[133,12782,155],{"stretchy":135},[133,12784,1403],{},[138,12786,825],{},[138,12788,1956],{},[133,12790,147],{},[138,12792,825],{},[138,12794,5648],{},[157,12796,12797],{"encoding":159},"rx = r(a + b) = ra + rb",[114,12799,12801,12822,12846,12867,12888],{"className":12800,"ariaHidden":143},[164],[114,12802,12804,12807,12810,12813,12816,12819],{"className":12803},[168],[114,12805],{"className":12806,"style":1420},[172],[114,12808,825],{"className":12809,"style":856},[181,182],[114,12811,835],{"className":12812},[181,182],[114,12814],{"className":12815,"style":250},[191],[114,12817,1403],{"className":12818},[254],[114,12820],{"className":12821,"style":250},[191],[114,12823,12825,12828,12831,12834,12837,12840,12843],{"className":12824},[168],[114,12826],{"className":12827,"style":173},[172],[114,12829,825],{"className":12830,"style":856},[181,182],[114,12832,136],{"className":12833},[177],[114,12835,1956],{"className":12836},[181,182],[114,12838],{"className":12839,"style":652},[191],[114,12841,147],{"className":12842},[656],[114,12844],{"className":12845,"style":652},[191],[114,12847,12849,12852,12855,12858,12861,12864],{"className":12848},[168],[114,12850],{"className":12851,"style":173},[172],[114,12853,5648],{"className":12854},[181,182],[114,12856,155],{"className":12857},[208],[114,12859],{"className":12860,"style":250},[191],[114,12862,1403],{"className":12863},[254],[114,12865],{"className":12866,"style":250},[191],[114,12868,12870,12873,12876,12879,12882,12885],{"className":12869},[168],[114,12871],{"className":12872,"style":10327},[172],[114,12874,825],{"className":12875,"style":856},[181,182],[114,12877,1956],{"className":12878},[181,182],[114,12880],{"className":12881,"style":652},[191],[114,12883,147],{"className":12884},[656],[114,12886],{"className":12887,"style":652},[191],[114,12889,12891,12894,12897],{"className":12890},[168],[114,12892],{"className":12893,"style":5300},[172],[114,12895,825],{"className":12896,"style":856},[181,182],[114,12898,5648],{"className":12899},[181,182],"。\n因 ",[114,12902,12904,12917],{"className":12903},[117],[114,12905,12907],{"className":12906},[121],[123,12908,12909],{"xmlns":125},[127,12910,12911,12915],{},[130,12912,12913],{},[138,12914,756],{},[157,12916,756],{"encoding":159},[114,12918,12920],{"className":12919,"ariaHidden":143},[164],[114,12921,12923,12926],{"className":12922},[168],[114,12924],{"className":12925,"style":264},[172],[114,12927,756],{"className":12928,"style":771},[181,182]," 是理想 ",[114,12931,12933,12959],{"className":12932},[117],[114,12934,12936],{"className":12935},[121],[123,12937,12938],{"xmlns":125},[127,12939,12940,12956],{},[130,12941,12942,12944,12946,12948,12950,12952,12954],{},[6955,12943,7061],{},[133,12945,10675],{},[6955,12947,7061],{},[138,12949,825],{},[138,12951,1956],{},[133,12953,828],{},[138,12955,756],{},[157,12957,12958],{"encoding":159},"\\implies ra \\in I",[114,12960,12962,12981,13002],{"className":12961,"ariaHidden":143},[164],[114,12963,12965,12969,12972,12975,12978],{"className":12964},[168],[114,12966],{"className":12967,"style":12968},[172],"height:0.549em;vertical-align:-0.024em;",[114,12970],{"className":12971,"style":250},[191],[114,12973,10675],{"className":12974},[254],[114,12976],{"className":12977,"style":250},[191],[114,12979],{"className":12980,"style":250},[191],[114,12982,12984,12987,12990,12993,12996,12999],{"className":12983},[168],[114,12985],{"className":12986,"style":852},[172],[114,12988,825],{"className":12989,"style":856},[181,182],[114,12991,1956],{"className":12992},[181,182],[114,12994],{"className":12995,"style":250},[191],[114,12997,828],{"className":12998},[254],[114,13000],{"className":13001,"style":250},[191],[114,13003,13005,13008],{"className":13004},[168],[114,13006],{"className":13007,"style":264},[172],[114,13009,756],{"className":13010,"style":771},[181,182],[114,13012,13014,13027],{"className":13013},[117],[114,13015,13017],{"className":13016},[121],[123,13018,13019],{"xmlns":125},[127,13020,13021,13025],{},[130,13022,13023],{},[138,13024,9856],{},[157,13026,9856],{"encoding":159},[114,13028,13030],{"className":13029,"ariaHidden":143},[164],[114,13031,13033,13036],{"className":13032},[168],[114,13034],{"className":13035,"style":264},[172],[114,13037,9856],{"className":13038,"style":9881},[181,182],[114,13040,13042,13068],{"className":13041},[117],[114,13043,13045],{"className":13044},[121],[123,13046,13047],{"xmlns":125},[127,13048,13049,13065],{},[130,13050,13051,13053,13055,13057,13059,13061,13063],{},[6955,13052,7061],{},[133,13054,10675],{},[6955,13056,7061],{},[138,13058,825],{},[138,13060,5648],{},[133,13062,828],{},[138,13064,9856],{},[157,13066,13067],{"encoding":159},"\\implies rb \\in J",[114,13069,13071,13089,13110],{"className":13070,"ariaHidden":143},[164],[114,13072,13074,13077,13080,13083,13086],{"className":13073},[168],[114,13075],{"className":13076,"style":12968},[172],[114,13078],{"className":13079,"style":250},[191],[114,13081,10675],{"className":13082},[254],[114,13084],{"className":13085,"style":250},[191],[114,13087],{"className":13088,"style":250},[191],[114,13090,13092,13095,13098,13101,13104,13107],{"className":13091},[168],[114,13093],{"className":13094,"style":10346},[172],[114,13096,825],{"className":13097,"style":856},[181,182],[114,13099,5648],{"className":13100},[181,182],[114,13102],{"className":13103,"style":250},[191],[114,13105,828],{"className":13106},[254],[114,13108],{"className":13109,"style":250},[191],[114,13111,13113,13116],{"className":13112},[168],[114,13114],{"className":13115,"style":264},[172],[114,13117,9856],{"className":13118,"style":9881},[181,182],"。\n所以 ",[114,13121,13123,13147],{"className":13122},[117],[114,13124,13126],{"className":13125},[121],[123,13127,13128],{"xmlns":125},[127,13129,13130,13144],{},[130,13131,13132,13134,13136,13138,13140,13142],{},[138,13133,825],{},[138,13135,835],{},[133,13137,828],{},[138,13139,756],{},[133,13141,147],{},[138,13143,9856],{},[157,13145,13146],{"encoding":159},"rx \\in I + J",[114,13148,13150,13171,13189],{"className":13149,"ariaHidden":143},[164],[114,13151,13153,13156,13159,13162,13165,13168],{"className":13152},[168],[114,13154],{"className":13155,"style":852},[172],[114,13157,825],{"className":13158,"style":856},[181,182],[114,13160,835],{"className":13161},[181,182],[114,13163],{"className":13164,"style":250},[191],[114,13166,828],{"className":13167},[254],[114,13169],{"className":13170,"style":250},[191],[114,13172,13174,13177,13180,13183,13186],{"className":13173},[168],[114,13175],{"className":13176,"style":11235},[172],[114,13178,756],{"className":13179,"style":771},[181,182],[114,13181],{"className":13182,"style":652},[191],[114,13184,147],{"className":13185},[656],[114,13187],{"className":13188,"style":652},[191],[114,13190,13192,13195],{"className":13191},[168],[114,13193],{"className":13194,"style":264},[172],[114,13196,9856],{"className":13197,"style":9881},[181,182],"。右乘同理。\n",[447,13200,13201],{},[450,13202,13203,536,13205,13255,13256,1562,13284,13312],{},[51,13204,11149],{},[114,13206,13208,13225],{"className":13207},[117],[114,13209,13211],{"className":13210},[121],[123,13212,13213],{"xmlns":125},[127,13214,13215,13223],{},[130,13216,13217,13219,13221],{},[138,13218,756],{},[133,13220,147],{},[138,13222,9856],{},[157,13224,11225],{"encoding":159},[114,13226,13228,13246],{"className":13227,"ariaHidden":143},[164],[114,13229,13231,13234,13237,13240,13243],{"className":13230},[168],[114,13232],{"className":13233,"style":11235},[172],[114,13235,756],{"className":13236,"style":771},[181,182],[114,13238],{"className":13239,"style":652},[191],[114,13241,147],{"className":13242},[656],[114,13244],{"className":13245,"style":652},[191],[114,13247,13249,13252],{"className":13248},[168],[114,13250],{"className":13251,"style":264},[172],[114,13253,9856],{"className":13254,"style":9881},[181,182]," 是理想（且是包含 ",[114,13257,13259,13272],{"className":13258},[117],[114,13260,13262],{"className":13261},[121],[123,13263,13264],{"xmlns":125},[127,13265,13266,13270],{},[130,13267,13268],{},[138,13269,756],{},[157,13271,756],{"encoding":159},[114,13273,13275],{"className":13274,"ariaHidden":143},[164],[114,13276,13278,13281],{"className":13277},[168],[114,13279],{"className":13280,"style":264},[172],[114,13282,756],{"className":13283,"style":771},[181,182],[114,13285,13287,13300],{"className":13286},[117],[114,13288,13290],{"className":13289},[121],[123,13291,13292],{"xmlns":125},[127,13293,13294,13298],{},[130,13295,13296],{},[138,13297,9856],{},[157,13299,9856],{"encoding":159},[114,13301,13303],{"className":13302,"ariaHidden":143},[164],[114,13304,13306,13309],{"className":13305},[168],[114,13307],{"className":13308,"style":264},[172],[114,13310,9856],{"className":13311,"style":9881},[181,182]," 的最小理想）。",[106,13314,13316,13317],{"id":13315},"_53-积-ijijij","5.3 积 ",[114,13318,13320,13336],{"className":13319},[117],[114,13321,13323],{"className":13322},[121],[123,13324,13325],{"xmlns":125},[127,13326,13327,13333],{},[130,13328,13329,13331],{},[138,13330,756],{},[138,13332,9856],{},[157,13334,13335],{"encoding":159},"IJ",[114,13337,13339],{"className":13338,"ariaHidden":143},[164],[114,13340,13342,13345,13348],{"className":13341},[168],[114,13343],{"className":13344,"style":264},[172],[114,13346,756],{"className":13347,"style":771},[181,182],[114,13349,9856],{"className":13350,"style":9881},[181,182],[19,13352,13353,536,13355,13787,13788,536,13790,10022,13823,536],{},[51,13354,11261],{},[114,13356,13358,13439],{"className":13357},[117],[114,13359,13361],{"className":13360},[121],[123,13362,13363],{"xmlns":125},[127,13364,13365,13436],{},[130,13366,13367,13369,13371,13373,13375,13389,13395,13401,13403,13409,13411,13413,13415,13421,13423,13425,13427,13429,13431,13434],{},[138,13368,756],{},[138,13370,9856],{},[133,13372,1403],{},[133,13374,626],{"stretchy":135},[6982,13376,13377,13379,13387],{},[133,13378,5631],{},[130,13380,13381,13383,13385],{},[138,13382,5288],{},[133,13384,1403],{},[550,13386,4373],{},[138,13388,1481],{},[1952,13390,13391,13393],{},[138,13392,1956],{},[138,13394,5288],{},[1952,13396,13397,13399],{},[138,13398,5648],{},[138,13400,5288],{},[133,13402,9740],{},[1952,13404,13405,13407],{},[138,13406,1956],{},[138,13408,5288],{},[133,13410,828],{},[138,13412,756],{},[133,13414,144],{"separator":143},[1952,13416,13417,13419],{},[138,13418,5648],{},[138,13420,5288],{},[133,13422,828],{},[138,13424,9856],{},[133,13426,144],{"separator":143},[138,13428,1481],{},[133,13430,828],{},[138,13432,13433],{"mathvariant":555},"N",[133,13435,632],{"stretchy":135},[157,13437,13438],{"encoding":159},"IJ = \\{ \\sum_{k=1}^n a_k b_k \\mid a_k \\in I, b_k \\in J, n \\in \\mathbb{N} \\}",[114,13440,13442,13463,13628,13684,13748,13775],{"className":13441,"ariaHidden":143},[164],[114,13443,13445,13448,13451,13454,13457,13460],{"className":13444},[168],[114,13446],{"className":13447,"style":264},[172],[114,13449,756],{"className":13450,"style":771},[181,182],[114,13452,9856],{"className":13453,"style":9881},[181,182],[114,13455],{"className":13456,"style":250},[191],[114,13458,1403],{"className":13459},[254],[114,13461],{"className":13462,"style":250},[191],[114,13464,13466,13470,13473,13536,13539,13579,13619,13622,13625],{"className":13465},[168],[114,13467],{"className":13468,"style":13469},[172],"height:1.104em;vertical-align:-0.2997em;",[114,13471,626],{"className":13472},[177],[114,13474,13476,13479],{"className":13475},[5732],[114,13477,5631],{"className":13478,"style":5738},[5732,5736,5737],[114,13480,13482],{"className":13481},[2010],[114,13483,13485,13527],{"className":13484},[2014,2015],[114,13486,13488,13524],{"className":13487},[2019],[114,13489,13492,13512],{"className":13490,"style":13491},[2023],"height:0.8043em;",[114,13493,13494,13497],{"style":5754},[114,13495],{"className":13496,"style":2032},[2031],[114,13498,13500],{"className":13499},[2036,2037,2038,2039],[114,13501,13503,13506,13509],{"className":13502},[181,2039],[114,13504,5288],{"className":13505,"style":5304},[181,182,2039],[114,13507,1403],{"className":13508},[254,2039],[114,13510,4373],{"className":13511},[181,2039],[114,13513,13515,13518],{"style":13514},"top:-3.2029em;margin-right:0.05em;",[114,13516],{"className":13517,"style":2032},[2031],[114,13519,13521],{"className":13520},[2036,2037,2038,2039],[114,13522,1481],{"className":13523},[181,182,2039],[114,13525,2052],{"className":13526},[2051],[114,13528,13530],{"className":13529},[2019],[114,13531,13534],{"className":13532,"style":13533},[2023],"height:0.2997em;",[114,13535],{},[114,13537],{"className":13538,"style":192},[191],[114,13540,13542,13545],{"className":13541},[181],[114,13543,1956],{"className":13544},[181,182],[114,13546,13548],{"className":13547},[2010],[114,13549,13551,13571],{"className":13550},[2014,2015],[114,13552,13554,13568],{"className":13553},[2019],[114,13555,13557],{"className":13556,"style":5686},[2023],[114,13558,13559,13562],{"style":2027},[114,13560],{"className":13561,"style":2032},[2031],[114,13563,13565],{"className":13564},[2036,2037,2038,2039],[114,13566,5288],{"className":13567,"style":5304},[181,182,2039],[114,13569,2052],{"className":13570},[2051],[114,13572,13574],{"className":13573},[2019],[114,13575,13577],{"className":13576,"style":2850},[2023],[114,13578],{},[114,13580,13582,13585],{"className":13581},[181],[114,13583,5648],{"className":13584},[181,182],[114,13586,13588],{"className":13587},[2010],[114,13589,13591,13611],{"className":13590},[2014,2015],[114,13592,13594,13608],{"className":13593},[2019],[114,13595,13597],{"className":13596,"style":5686},[2023],[114,13598,13599,13602],{"style":2027},[114,13600],{"className":13601,"style":2032},[2031],[114,13603,13605],{"className":13604},[2036,2037,2038,2039],[114,13606,5288],{"className":13607,"style":5304},[181,182,2039],[114,13609,2052],{"className":13610},[2051],[114,13612,13614],{"className":13613},[2019],[114,13615,13617],{"className":13616,"style":2850},[2023],[114,13618],{},[114,13620],{"className":13621,"style":250},[191],[114,13623,9740],{"className":13624},[254],[114,13626],{"className":13627,"style":250},[191],[114,13629,13631,13635,13675,13678,13681],{"className":13630},[168],[114,13632],{"className":13633,"style":13634},[172],"height:0.6891em;vertical-align:-0.15em;",[114,13636,13638,13641],{"className":13637},[181],[114,13639,1956],{"className":13640},[181,182],[114,13642,13644],{"className":13643},[2010],[114,13645,13647,13667],{"className":13646},[2014,2015],[114,13648,13650,13664],{"className":13649},[2019],[114,13651,13653],{"className":13652,"style":5686},[2023],[114,13654,13655,13658],{"style":2027},[114,13656],{"className":13657,"style":2032},[2031],[114,13659,13661],{"className":13660},[2036,2037,2038,2039],[114,13662,5288],{"className":13663,"style":5304},[181,182,2039],[114,13665,2052],{"className":13666},[2051],[114,13668,13670],{"className":13669},[2019],[114,13671,13673],{"className":13672,"style":2850},[2023],[114,13674],{},[114,13676],{"className":13677,"style":250},[191],[114,13679,828],{"className":13680},[254],[114,13682],{"className":13683,"style":250},[191],[114,13685,13687,13690,13693,13696,13699,13739,13742,13745],{"className":13686},[168],[114,13688],{"className":13689,"style":1810},[172],[114,13691,756],{"className":13692,"style":771},[181,182],[114,13694,144],{"className":13695},[187],[114,13697],{"className":13698,"style":192},[191],[114,13700,13702,13705],{"className":13701},[181],[114,13703,5648],{"className":13704},[181,182],[114,13706,13708],{"className":13707},[2010],[114,13709,13711,13731],{"className":13710},[2014,2015],[114,13712,13714,13728],{"className":13713},[2019],[114,13715,13717],{"className":13716,"style":5686},[2023],[114,13718,13719,13722],{"style":2027},[114,13720],{"className":13721,"style":2032},[2031],[114,13723,13725],{"className":13724},[2036,2037,2038,2039],[114,13726,5288],{"className":13727,"style":5304},[181,182,2039],[114,13729,2052],{"className":13730},[2051],[114,13732,13734],{"className":13733},[2019],[114,13735,13737],{"className":13736,"style":2850},[2023],[114,13738],{},[114,13740],{"className":13741,"style":250},[191],[114,13743,828],{"className":13744},[254],[114,13746],{"className":13747,"style":250},[191],[114,13749,13751,13754,13757,13760,13763,13766,13769,13772],{"className":13750},[168],[114,13752],{"className":13753,"style":872},[172],[114,13755,9856],{"className":13756,"style":9881},[181,182],[114,13758,144],{"className":13759},[187],[114,13761],{"className":13762,"style":192},[191],[114,13764,1481],{"className":13765},[181,182],[114,13767],{"className":13768,"style":250},[191],[114,13770,828],{"className":13771},[254],[114,13773],{"className":13774,"style":250},[191],[114,13776,13778,13781,13784],{"className":13777},[168],[114,13779],{"className":13780,"style":173},[172],[114,13782,13433],{"className":13783},[181,576],[114,13785,632],{"className":13786},[208],"（注意是有限和形式）。\n",[51,13789,9971],{},[114,13791,13793,13808],{"className":13792},[117],[114,13794,13796],{"className":13795},[121],[123,13797,13798],{"xmlns":125},[127,13799,13800,13806],{},[130,13801,13802,13804],{},[138,13803,756],{},[138,13805,9856],{},[157,13807,13335],{"encoding":159},[114,13809,13811],{"className":13810,"ariaHidden":143},[164],[114,13812,13814,13817,13820],{"className":13813},[168],[114,13815],{"className":13816,"style":264},[172],[114,13818,756],{"className":13819,"style":771},[181,182],[114,13821,9856],{"className":13822,"style":9881},[181,182],[51,13824,10025],{},[803,13826,13827,13866],{},[450,13828,13829,13831,13832,13865],{},[51,13830,10032],{},"：由定义知，",[114,13833,13835,13850],{"className":13834},[117],[114,13836,13838],{"className":13837},[121],[123,13839,13840],{"xmlns":125},[127,13841,13842,13848],{},[130,13843,13844,13846],{},[138,13845,756],{},[138,13847,9856],{},[157,13849,13335],{"encoding":159},[114,13851,13853],{"className":13852,"ariaHidden":143},[164],[114,13854,13856,13859,13862],{"className":13855},[168],[114,13857],{"className":13858,"style":264},[172],[114,13860,756],{"className":13861,"style":771},[181,182],[114,13863,9856],{"className":13864,"style":9881},[181,182]," 对加法和减法封闭（因为它是有限和的集合）。",[450,13867,13868,13870,13871,13902,13903,13956,13957,14043,14044,1595,14094],{},[51,13869,10540],{},"：只需对单个项 ",[114,13872,13874,13890],{"className":13873},[117],[114,13875,13877],{"className":13876},[121],[123,13878,13879],{"xmlns":125},[127,13880,13881,13887],{},[130,13882,13883,13885],{},[138,13884,1956],{},[138,13886,5648],{},[157,13888,13889],{"encoding":159},"ab",[114,13891,13893],{"className":13892,"ariaHidden":143},[164],[114,13894,13896,13899],{"className":13895},[168],[114,13897],{"className":13898,"style":5300},[172],[114,13900,13889],{"className":13901},[181,182]," 验证，有限和自然满足。\n设 ",[114,13904,13906,13926],{"className":13905},[117],[114,13907,13909],{"className":13908},[121],[123,13910,13911],{"xmlns":125},[127,13912,13913,13923],{},[130,13914,13915,13917,13919,13921],{},[138,13916,835],{},[133,13918,1403],{},[138,13920,1956],{},[138,13922,5648],{},[157,13924,13925],{"encoding":159},"x = ab",[114,13927,13929,13947],{"className":13928,"ariaHidden":143},[164],[114,13930,13932,13935,13938,13941,13944],{"className":13931},[168],[114,13933],{"className":13934,"style":1420},[172],[114,13936,835],{"className":13937},[181,182],[114,13939],{"className":13940,"style":250},[191],[114,13942,1403],{"className":13943},[254],[114,13945],{"className":13946,"style":250},[191],[114,13948,13950,13953],{"className":13949},[168],[114,13951],{"className":13952,"style":5300},[172],[114,13954,13889],{"className":13955},[181,182],"，其中 ",[114,13958,13960,13986],{"className":13959},[117],[114,13961,13963],{"className":13962},[121],[123,13964,13965],{"xmlns":125},[127,13966,13967,13983],{},[130,13968,13969,13971,13973,13975,13977,13979,13981],{},[138,13970,1956],{},[133,13972,828],{},[138,13974,756],{},[133,13976,144],{"separator":143},[138,13978,5648],{},[133,13980,828],{},[138,13982,9856],{},[157,13984,13985],{"encoding":159},"a \\in I, b \\in J",[114,13987,13989,14007,14034],{"className":13988,"ariaHidden":143},[164],[114,13990,13992,13995,13998,14001,14004],{"className":13991},[168],[114,13993],{"className":13994,"style":852},[172],[114,13996,1956],{"className":13997},[181,182],[114,13999],{"className":14000,"style":250},[191],[114,14002,828],{"className":14003},[254],[114,14005],{"className":14006,"style":250},[191],[114,14008,14010,14013,14016,14019,14022,14025,14028,14031],{"className":14009},[168],[114,14011],{"className":14012,"style":1810},[172],[114,14014,756],{"className":14015,"style":771},[181,182],[114,14017,144],{"className":14018},[187],[114,14020],{"className":14021,"style":192},[191],[114,14023,5648],{"className":14024},[181,182],[114,14026],{"className":14027,"style":250},[191],[114,14029,828],{"className":14030},[254],[114,14032],{"className":14033,"style":250},[191],[114,14035,14037,14040],{"className":14036},[168],[114,14038],{"className":14039,"style":264},[172],[114,14041,9856],{"className":14042,"style":9881},[181,182],"。设 ",[114,14045,14047,14064],{"className":14046},[117],[114,14048,14050],{"className":14049},[121],[123,14051,14052],{"xmlns":125},[127,14053,14054,14062],{},[130,14055,14056,14058,14060],{},[138,14057,825],{},[133,14059,828],{},[138,14061,140],{},[157,14063,12722],{"encoding":159},[114,14065,14067,14085],{"className":14066,"ariaHidden":143},[164],[114,14068,14070,14073,14076,14079,14082],{"className":14069},[168],[114,14071],{"className":14072,"style":852},[172],[114,14074,825],{"className":14075,"style":856},[181,182],[114,14077],{"className":14078,"style":250},[191],[114,14080,828],{"className":14081},[254],[114,14083],{"className":14084,"style":250},[191],[114,14086,14088,14091],{"className":14087},[168],[114,14089],{"className":14090,"style":264},[172],[114,14092,140],{"className":14093,"style":183},[181,182],[447,14095,14096,14383,14668,14705],{},[450,14097,14098,14213,14214,14242,14243,14299,14300,14349,14350,445],{},[114,14099,14101,14141],{"className":14100},[117],[114,14102,14104],{"className":14103},[121],[123,14105,14106],{"xmlns":125},[127,14107,14108,14138],{},[130,14109,14110,14112,14114,14116,14118,14120,14122,14124,14126,14128,14130,14132,14134,14136],{},[138,14111,825],{},[138,14113,835],{},[133,14115,1403],{},[138,14117,825],{},[133,14119,136],{"stretchy":135},[138,14121,1956],{},[138,14123,5648],{},[133,14125,155],{"stretchy":135},[133,14127,1403],{},[133,14129,136],{"stretchy":135},[138,14131,825],{},[138,14133,1956],{},[133,14135,155],{"stretchy":135},[138,14137,5648],{},[157,14139,14140],{"encoding":159},"r x = r (ab) = (ra) b",[114,14142,14144,14165,14192],{"className":14143,"ariaHidden":143},[164],[114,14145,14147,14150,14153,14156,14159,14162],{"className":14146},[168],[114,14148],{"className":14149,"style":1420},[172],[114,14151,825],{"className":14152,"style":856},[181,182],[114,14154,835],{"className":14155},[181,182],[114,14157],{"className":14158,"style":250},[191],[114,14160,1403],{"className":14161},[254],[114,14163],{"className":14164,"style":250},[191],[114,14166,14168,14171,14174,14177,14180,14183,14186,14189],{"className":14167},[168],[114,14169],{"className":14170,"style":173},[172],[114,14172,825],{"className":14173,"style":856},[181,182],[114,14175,136],{"className":14176},[177],[114,14178,13889],{"className":14179},[181,182],[114,14181,155],{"className":14182},[208],[114,14184],{"className":14185,"style":250},[191],[114,14187,1403],{"className":14188},[254],[114,14190],{"className":14191,"style":250},[191],[114,14193,14195,14198,14201,14204,14207,14210],{"className":14194},[168],[114,14196],{"className":14197,"style":173},[172],[114,14199,136],{"className":14200},[177],[114,14202,825],{"className":14203,"style":856},[181,182],[114,14205,1956],{"className":14206},[181,182],[114,14208,155],{"className":14209},[208],[114,14211,5648],{"className":14212},[181,182],"。因为 ",[114,14215,14217,14230],{"className":14216},[117],[114,14218,14220],{"className":14219},[121],[123,14221,14222],{"xmlns":125},[127,14223,14224,14228],{},[130,14225,14226],{},[138,14227,756],{},[157,14229,756],{"encoding":159},[114,14231,14233],{"className":14232,"ariaHidden":143},[164],[114,14234,14236,14239],{"className":14235},[168],[114,14237],{"className":14238,"style":264},[172],[114,14240,756],{"className":14241,"style":771},[181,182]," 是左理想，",[114,14244,14246,14266],{"className":14245},[117],[114,14247,14249],{"className":14248},[121],[123,14250,14251],{"xmlns":125},[127,14252,14253,14263],{},[130,14254,14255,14257,14259,14261],{},[138,14256,825],{},[138,14258,1956],{},[133,14260,828],{},[138,14262,756],{},[157,14264,14265],{"encoding":159},"ra \\in I",[114,14267,14269,14290],{"className":14268,"ariaHidden":143},[164],[114,14270,14272,14275,14278,14281,14284,14287],{"className":14271},[168],[114,14273],{"className":14274,"style":852},[172],[114,14276,825],{"className":14277,"style":856},[181,182],[114,14279,1956],{"className":14280},[181,182],[114,14282],{"className":14283,"style":250},[191],[114,14285,828],{"className":14286},[254],[114,14288],{"className":14289,"style":250},[191],[114,14291,14293,14296],{"className":14292},[168],[114,14294],{"className":14295,"style":264},[172],[114,14297,756],{"className":14298,"style":771},[181,182],"。所以 ",[114,14301,14303,14325],{"className":14302},[117],[114,14304,14306],{"className":14305},[121],[123,14307,14308],{"xmlns":125},[127,14309,14310,14322],{},[130,14311,14312,14314,14316,14318,14320],{},[133,14313,136],{"stretchy":135},[138,14315,825],{},[138,14317,1956],{},[133,14319,155],{"stretchy":135},[138,14321,5648],{},[157,14323,14324],{"encoding":159},"(ra)b",[114,14326,14328],{"className":14327,"ariaHidden":143},[164],[114,14329,14331,14334,14337,14340,14343,14346],{"className":14330},[168],[114,14332],{"className":14333,"style":173},[172],[114,14335,136],{"className":14336},[177],[114,14338,825],{"className":14339,"style":856},[181,182],[114,14341,1956],{"className":14342},[181,182],[114,14344,155],{"className":14345},[208],[114,14347,5648],{"className":14348},[181,182]," 仍是形如 \"(I中元素)(J中元素)\" 的形式，属于 ",[114,14351,14353,14368],{"className":14352},[117],[114,14354,14356],{"className":14355},[121],[123,14357,14358],{"xmlns":125},[127,14359,14360,14366],{},[130,14361,14362,14364],{},[138,14363,756],{},[138,14365,9856],{},[157,14367,13335],{"encoding":159},[114,14369,14371],{"className":14370,"ariaHidden":143},[164],[114,14372,14374,14377,14380],{"className":14373},[168],[114,14375],{"className":14376,"style":264},[172],[114,14378,756],{"className":14379,"style":771},[181,182],[114,14381,9856],{"className":14382,"style":9881},[181,182],[450,14384,14385,14213,14500,14528,14529,14299,14585,14634,14635,445],{},[114,14386,14388,14428],{"className":14387},[117],[114,14389,14391],{"className":14390},[121],[123,14392,14393],{"xmlns":125},[127,14394,14395,14425],{},[130,14396,14397,14399,14401,14403,14405,14407,14409,14411,14413,14415,14417,14419,14421,14423],{},[138,14398,835],{},[138,14400,825],{},[133,14402,1403],{},[133,14404,136],{"stretchy":135},[138,14406,1956],{},[138,14408,5648],{},[133,14410,155],{"stretchy":135},[138,14412,825],{},[133,14414,1403],{},[138,14416,1956],{},[133,14418,136],{"stretchy":135},[138,14420,5648],{},[138,14422,825],{},[133,14424,155],{"stretchy":135},[157,14426,14427],{"encoding":159},"x r = (ab) r = a (br)",[114,14429,14431,14452,14479],{"className":14430,"ariaHidden":143},[164],[114,14432,14434,14437,14440,14443,14446,14449],{"className":14433},[168],[114,14435],{"className":14436,"style":1420},[172],[114,14438,835],{"className":14439},[181,182],[114,14441,825],{"className":14442,"style":856},[181,182],[114,14444],{"className":14445,"style":250},[191],[114,14447,1403],{"className":14448},[254],[114,14450],{"className":14451,"style":250},[191],[114,14453,14455,14458,14461,14464,14467,14470,14473,14476],{"className":14454},[168],[114,14456],{"className":14457,"style":173},[172],[114,14459,136],{"className":14460},[177],[114,14462,13889],{"className":14463},[181,182],[114,14465,155],{"className":14466},[208],[114,14468,825],{"className":14469,"style":856},[181,182],[114,14471],{"className":14472,"style":250},[191],[114,14474,1403],{"className":14475},[254],[114,14477],{"className":14478,"style":250},[191],[114,14480,14482,14485,14488,14491,14494,14497],{"className":14481},[168],[114,14483],{"className":14484,"style":173},[172],[114,14486,1956],{"className":14487},[181,182],[114,14489,136],{"className":14490},[177],[114,14492,5648],{"className":14493},[181,182],[114,14495,825],{"className":14496,"style":856},[181,182],[114,14498,155],{"className":14499},[208],[114,14501,14503,14516],{"className":14502},[117],[114,14504,14506],{"className":14505},[121],[123,14507,14508],{"xmlns":125},[127,14509,14510,14514],{},[130,14511,14512],{},[138,14513,9856],{},[157,14515,9856],{"encoding":159},[114,14517,14519],{"className":14518,"ariaHidden":143},[164],[114,14520,14522,14525],{"className":14521},[168],[114,14523],{"className":14524,"style":264},[172],[114,14526,9856],{"className":14527,"style":9881},[181,182]," 是右理想，",[114,14530,14532,14552],{"className":14531},[117],[114,14533,14535],{"className":14534},[121],[123,14536,14537],{"xmlns":125},[127,14538,14539,14549],{},[130,14540,14541,14543,14545,14547],{},[138,14542,5648],{},[138,14544,825],{},[133,14546,828],{},[138,14548,9856],{},[157,14550,14551],{"encoding":159},"br \\in J",[114,14553,14555,14576],{"className":14554,"ariaHidden":143},[164],[114,14556,14558,14561,14564,14567,14570,14573],{"className":14557},[168],[114,14559],{"className":14560,"style":10346},[172],[114,14562,5648],{"className":14563},[181,182],[114,14565,825],{"className":14566,"style":856},[181,182],[114,14568],{"className":14569,"style":250},[191],[114,14571,828],{"className":14572},[254],[114,14574],{"className":14575,"style":250},[191],[114,14577,14579,14582],{"className":14578},[168],[114,14580],{"className":14581,"style":264},[172],[114,14583,9856],{"className":14584,"style":9881},[181,182],[114,14586,14588,14610],{"className":14587},[117],[114,14589,14591],{"className":14590},[121],[123,14592,14593],{"xmlns":125},[127,14594,14595,14607],{},[130,14596,14597,14599,14601,14603,14605],{},[138,14598,1956],{},[133,14600,136],{"stretchy":135},[138,14602,5648],{},[138,14604,825],{},[133,14606,155],{"stretchy":135},[157,14608,14609],{"encoding":159},"a(br)",[114,14611,14613],{"className":14612,"ariaHidden":143},[164],[114,14614,14616,14619,14622,14625,14628,14631],{"className":14615},[168],[114,14617],{"className":14618,"style":173},[172],[114,14620,1956],{"className":14621},[181,182],[114,14623,136],{"className":14624},[177],[114,14626,5648],{"className":14627},[181,182],[114,14629,825],{"className":14630,"style":856},[181,182],[114,14632,155],{"className":14633},[208]," 仍属于 ",[114,14636,14638,14653],{"className":14637},[117],[114,14639,14641],{"className":14640},[121],[123,14642,14643],{"xmlns":125},[127,14644,14645,14651],{},[130,14646,14647,14649],{},[138,14648,756],{},[138,14650,9856],{},[157,14652,13335],{"encoding":159},[114,14654,14656],{"className":14655,"ariaHidden":143},[164],[114,14657,14659,14662,14665],{"className":14658},[168],[114,14660],{"className":14661,"style":264},[172],[114,14663,756],{"className":14664,"style":771},[181,182],[114,14666,9856],{"className":14667,"style":9881},[181,182],[450,14669,14670,536,14672,11200],{},[51,14671,11149],{},[114,14673,14675,14690],{"className":14674},[117],[114,14676,14678],{"className":14677},[121],[123,14679,14680],{"xmlns":125},[127,14681,14682,14688],{},[130,14683,14684,14686],{},[138,14685,756],{},[138,14687,9856],{},[157,14689,13335],{"encoding":159},[114,14691,14693],{"className":14692,"ariaHidden":143},[164],[114,14694,14696,14699,14702],{"className":14695},[168],[114,14697],{"className":14698,"style":264},[172],[114,14700,756],{"className":14701,"style":771},[181,182],[114,14703,9856],{"className":14704,"style":9881},[181,182],[450,14706,14707,14710,14711,445],{},[533,14708,14709],{},"注意","：一般情况下 ",[114,14712,14714,14738],{"className":14713},[117],[114,14715,14717],{"className":14716},[121],[123,14718,14719],{"xmlns":125},[127,14720,14721,14735],{},[130,14722,14723,14725,14727,14729,14731,14733],{},[138,14724,756],{},[138,14726,9856],{},[133,14728,227],{},[138,14730,756],{},[133,14732,9931],{},[138,14734,9856],{},[157,14736,14737],{"encoding":159},"IJ \\subseteq I \\cap J",[114,14739,14741,14762,14780],{"className":14740,"ariaHidden":143},[164],[114,14742,14744,14747,14750,14753,14756,14759],{"className":14743},[168],[114,14745],{"className":14746,"style":242},[172],[114,14748,756],{"className":14749,"style":771},[181,182],[114,14751,9856],{"className":14752,"style":9881},[181,182],[114,14754],{"className":14755,"style":250},[191],[114,14757,227],{"className":14758},[254],[114,14760],{"className":14761,"style":250},[191],[114,14763,14765,14768,14771,14774,14777],{"className":14764},[168],[114,14766],{"className":14767,"style":264},[172],[114,14769,756],{"className":14770,"style":771},[181,182],[114,14772],{"className":14773,"style":652},[191],[114,14775,9931],{"className":14776},[656],[114,14778],{"className":14779,"style":652},[191],[114,14781,14783,14786],{"className":14782},[168],[114,14784],{"className":14785,"style":264},[172],[114,14787,9856],{"className":14788,"style":9881},[181,182],[99,14790],{},[14,14792,14794],{"id":14793},"_6-商环-quotient-ring","6. 商环 (Quotient Ring)",[106,14796,14798],{"id":14797},"_61-构造与良定性","6.1 构造与良定性",[19,14800,14801,14802,14830,14831,14833,14834,14837],{},"我们需要 ",[114,14803,14805,14818],{"className":14804},[117],[114,14806,14808],{"className":14807},[121],[123,14809,14810],{"xmlns":125},[127,14811,14812,14816],{},[130,14813,14814],{},[138,14815,756],{},[157,14817,756],{"encoding":159},[114,14819,14821],{"className":14820,"ariaHidden":143},[164],[114,14822,14824,14827],{"className":14823},[168],[114,14825],{"className":14826,"style":264},[172],[114,14828,756],{"className":14829,"style":771},[181,182]," 是",[51,14832,76],{},"（而不仅仅是子环），是为了保证商环上的乘法运算是",[51,14835,14836],{},"良定 (well-defined)"," 的。",[447,14839,14840,14984,15179],{},[450,14841,14842,536,14845,14983],{},[51,14843,14844],{},"集合",[114,14846,14848,14887],{"className":14847},[117],[114,14849,14851],{"className":14850},[121],[123,14852,14853],{"xmlns":125},[127,14854,14855,14884],{},[130,14856,14857,14859,14862,14864,14866,14868,14870,14872,14874,14876,14878,14880,14882],{},[138,14858,140],{},[138,14860,14861],{"mathvariant":1790},"\u002F",[138,14863,756],{},[133,14865,1403],{},[133,14867,626],{"stretchy":135},[138,14869,825],{},[133,14871,147],{},[138,14873,756],{},[133,14875,9740],{},[138,14877,825],{},[133,14879,828],{},[138,14881,140],{},[133,14883,632],{"stretchy":135},[157,14885,14886],{"encoding":159},"R\u002FI = \\{ r + I \\mid r \\in R \\}",[114,14888,14890,14914,14935,14953,14971],{"className":14889,"ariaHidden":143},[164],[114,14891,14893,14896,14899,14902,14905,14908,14911],{"className":14892},[168],[114,14894],{"className":14895,"style":173},[172],[114,14897,140],{"className":14898,"style":183},[181,182],[114,14900,14861],{"className":14901},[181],[114,14903,756],{"className":14904,"style":771},[181,182],[114,14906],{"className":14907,"style":250},[191],[114,14909,1403],{"className":14910},[254],[114,14912],{"className":14913,"style":250},[191],[114,14915,14917,14920,14923,14926,14929,14932],{"className":14916},[168],[114,14918],{"className":14919,"style":173},[172],[114,14921,626],{"className":14922},[177],[114,14924,825],{"className":14925,"style":856},[181,182],[114,14927],{"className":14928,"style":652},[191],[114,14930,147],{"className":14931},[656],[114,14933],{"className":14934,"style":652},[191],[114,14936,14938,14941,14944,14947,14950],{"className":14937},[168],[114,14939],{"className":14940,"style":173},[172],[114,14942,756],{"className":14943,"style":771},[181,182],[114,14945],{"className":14946,"style":250},[191],[114,14948,9740],{"className":14949},[254],[114,14951],{"className":14952,"style":250},[191],[114,14954,14956,14959,14962,14965,14968],{"className":14955},[168],[114,14957],{"className":14958,"style":852},[172],[114,14960,825],{"className":14961,"style":856},[181,182],[114,14963],{"className":14964,"style":250},[191],[114,14966,828],{"className":14967},[254],[114,14969],{"className":14970,"style":250},[191],[114,14972,14974,14977,14980],{"className":14973},[168],[114,14975],{"className":14976,"style":173},[172],[114,14978,140],{"className":14979,"style":183},[181,182],[114,14981,632],{"className":14982},[208],"（陪集）。",[450,14985,14986,536,14988,445],{},[51,14987,329],{},[114,14989,14991,15041],{"className":14990},[117],[114,14992,14994],{"className":14993},[121],[123,14995,14996],{"xmlns":125},[127,14997,14998,15038],{},[130,14999,15000,15002,15004,15006,15008,15010,15012,15014,15016,15018,15020,15022,15024,15026,15028,15030,15032,15034,15036],{},[133,15001,136],{"stretchy":135},[138,15003,1956],{},[133,15005,147],{},[138,15007,756],{},[133,15009,155],{"stretchy":135},[133,15011,147],{},[133,15013,136],{"stretchy":135},[138,15015,5648],{},[133,15017,147],{},[138,15019,756],{},[133,15021,155],{"stretchy":135},[133,15023,1403],{},[133,15025,136],{"stretchy":135},[138,15027,1956],{},[133,15029,147],{},[138,15031,5648],{},[133,15033,155],{"stretchy":135},[133,15035,147],{},[138,15037,756],{},[157,15039,15040],{"encoding":159},"(a + I) + (b + I) = (a + b) + I",[114,15042,15044,15065,15086,15107,15128,15149,15170],{"className":15043,"ariaHidden":143},[164],[114,15045,15047,15050,15053,15056,15059,15062],{"className":15046},[168],[114,15048],{"className":15049,"style":173},[172],[114,15051,136],{"className":15052},[177],[114,15054,1956],{"className":15055},[181,182],[114,15057],{"className":15058,"style":652},[191],[114,15060,147],{"className":15061},[656],[114,15063],{"className":15064,"style":652},[191],[114,15066,15068,15071,15074,15077,15080,15083],{"className":15067},[168],[114,15069],{"className":15070,"style":173},[172],[114,15072,756],{"className":15073,"style":771},[181,182],[114,15075,155],{"className":15076},[208],[114,15078],{"className":15079,"style":652},[191],[114,15081,147],{"className":15082},[656],[114,15084],{"className":15085,"style":652},[191],[114,15087,15089,15092,15095,15098,15101,15104],{"className":15088},[168],[114,15090],{"className":15091,"style":173},[172],[114,15093,136],{"className":15094},[177],[114,15096,5648],{"className":15097},[181,182],[114,15099],{"className":15100,"style":652},[191],[114,15102,147],{"className":15103},[656],[114,15105],{"className":15106,"style":652},[191],[114,15108,15110,15113,15116,15119,15122,15125],{"className":15109},[168],[114,15111],{"className":15112,"style":173},[172],[114,15114,756],{"className":15115,"style":771},[181,182],[114,15117,155],{"className":15118},[208],[114,15120],{"className":15121,"style":250},[191],[114,15123,1403],{"className":15124},[254],[114,15126],{"className":15127,"style":250},[191],[114,15129,15131,15134,15137,15140,15143,15146],{"className":15130},[168],[114,15132],{"className":15133,"style":173},[172],[114,15135,136],{"className":15136},[177],[114,15138,1956],{"className":15139},[181,182],[114,15141],{"className":15142,"style":652},[191],[114,15144,147],{"className":15145},[656],[114,15147],{"className":15148,"style":652},[191],[114,15150,15152,15155,15158,15161,15164,15167],{"className":15151},[168],[114,15153],{"className":15154,"style":173},[172],[114,15156,5648],{"className":15157},[181,182],[114,15159,155],{"className":15160},[208],[114,15162],{"className":15163,"style":652},[191],[114,15165,147],{"className":15166},[656],[114,15168],{"className":15169,"style":652},[191],[114,15171,15173,15176],{"className":15172},[168],[114,15174],{"className":15175,"style":264},[172],[114,15177,756],{"className":15178,"style":771},[181,182],[450,15180,15181,536,15183,445],{},[51,15182,333],{},[114,15184,15186,15228],{"className":15185},[117],[114,15187,15189],{"className":15188},[121],[123,15190,15191],{"xmlns":125},[127,15192,15193,15225],{},[130,15194,15195,15197,15199,15201,15203,15205,15207,15209,15211,15213,15215,15217,15219,15221,15223],{},[133,15196,136],{"stretchy":135},[138,15198,1956],{},[133,15200,147],{},[138,15202,756],{},[133,15204,155],{"stretchy":135},[133,15206,136],{"stretchy":135},[138,15208,5648],{},[133,15210,147],{},[138,15212,756],{},[133,15214,155],{"stretchy":135},[133,15216,1403],{},[138,15218,1956],{},[138,15220,5648],{},[133,15222,147],{},[138,15224,756],{},[157,15226,15227],{"encoding":159},"(a + I)(b + I) = ab + I",[114,15229,15231,15252,15279,15300,15319],{"className":15230,"ariaHidden":143},[164],[114,15232,15234,15237,15240,15243,15246,15249],{"className":15233},[168],[114,15235],{"className":15236,"style":173},[172],[114,15238,136],{"className":15239},[177],[114,15241,1956],{"className":15242},[181,182],[114,15244],{"className":15245,"style":652},[191],[114,15247,147],{"className":15248},[656],[114,15250],{"className":15251,"style":652},[191],[114,15253,15255,15258,15261,15264,15267,15270,15273,15276],{"className":15254},[168],[114,15256],{"className":15257,"style":173},[172],[114,15259,756],{"className":15260,"style":771},[181,182],[114,15262,155],{"className":15263},[208],[114,15265,136],{"className":15266},[177],[114,15268,5648],{"className":15269},[181,182],[114,15271],{"className":15272,"style":652},[191],[114,15274,147],{"className":15275},[656],[114,15277],{"className":15278,"style":652},[191],[114,15280,15282,15285,15288,15291,15294,15297],{"className":15281},[168],[114,15283],{"className":15284,"style":173},[172],[114,15286,756],{"className":15287,"style":771},[181,182],[114,15289,155],{"className":15290},[208],[114,15292],{"className":15293,"style":250},[191],[114,15295,1403],{"className":15296},[254],[114,15298],{"className":15299,"style":250},[191],[114,15301,15303,15307,15310,15313,15316],{"className":15302},[168],[114,15304],{"className":15305,"style":15306},[172],"height:0.7778em;vertical-align:-0.0833em;",[114,15308,13889],{"className":15309},[181,182],[114,15311],{"className":15312,"style":652},[191],[114,15314,147],{"className":15315},[656],[114,15317],{"className":15318,"style":652},[191],[114,15320,15322,15325],{"className":15321},[168],[114,15323],{"className":15324,"style":264},[172],[114,15326,756],{"className":15327,"style":771},[181,182],[19,15329,15330,15333,15334,15532,15533,15729,15730,15794,15795,16537,16538,16566],{},[51,15331,15332],{},"乘法良定性证明","：\n若 ",[114,15335,15337,15379],{"className":15336},[117],[114,15338,15340],{"className":15339},[121],[123,15341,15342],{"xmlns":125},[127,15343,15344,15376],{},[130,15345,15346,15352,15354,15356,15358,15360,15362,15368,15370,15372,15374],{},[8598,15347,15348,15350],{},[138,15349,1956],{},[133,15351,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[133,15353,828],{},[138,15355,1956],{},[133,15357,147],{},[138,15359,756],{},[133,15361,144],{"separator":143},[8598,15363,15364,15366],{},[138,15365,5648],{},[133,15367,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[133,15369,828],{},[138,15371,5648],{},[133,15373,147],{},[138,15375,756],{},[157,15377,15378],{"encoding":159},"a' \\in a+I, b' \\in b+I",[114,15380,15382,15430,15448,15505,15523],{"className":15381,"ariaHidden":143},[164],[114,15383,15385,15389,15421,15424,15427],{"className":15384},[168],[114,15386],{"className":15387,"style":15388},[172],"height:0.791em;vertical-align:-0.0391em;",[114,15390,15392,15395],{"className":15391},[181],[114,15393,1956],{"className":15394},[181,182],[114,15396,15398],{"className":15397},[2010],[114,15399,15401],{"className":15400},[2014],[114,15402,15404],{"className":15403},[2019],[114,15405,15407],{"className":15406,"style":8192},[2023],[114,15408,15409,15412],{"style":8207},[114,15410],{"className":15411,"style":2032},[2031],[114,15413,15415],{"className":15414},[2036,2037,2038,2039],[114,15416,15418],{"className":15417},[181,2039],[114,15419,6991],{"className":15420},[181,2039],[114,15422],{"className":15423,"style":250},[191],[114,15425,828],{"className":15426},[254],[114,15428],{"className":15429,"style":250},[191],[114,15431,15433,15436,15439,15442,15445],{"className":15432},[168],[114,15434],{"className":15435,"style":10327},[172],[114,15437,1956],{"className":15438},[181,182],[114,15440],{"className":15441,"style":652},[191],[114,15443,147],{"className":15444},[656],[114,15446],{"className":15447,"style":652},[191],[114,15449,15451,15455,15458,15461,15464,15496,15499,15502],{"className":15450},[168],[114,15452],{"className":15453,"style":15454},[172],"height:0.9463em;vertical-align:-0.1944em;",[114,15456,756],{"className":15457,"style":771},[181,182],[114,15459,144],{"className":15460},[187],[114,15462],{"className":15463,"style":192},[191],[114,15465,15467,15470],{"className":15466},[181],[114,15468,5648],{"className":15469},[181,182],[114,15471,15473],{"className":15472},[2010],[114,15474,15476],{"className":15475},[2014],[114,15477,15479],{"className":15478},[2019],[114,15480,15482],{"className":15481,"style":8192},[2023],[114,15483,15484,15487],{"style":8207},[114,15485],{"className":15486,"style":2032},[2031],[114,15488,15490],{"className":15489},[2036,2037,2038,2039],[114,15491,15493],{"className":15492},[181,2039],[114,15494,6991],{"className":15495},[181,2039],[114,15497],{"className":15498,"style":250},[191],[114,15500,828],{"className":15501},[254],[114,15503],{"className":15504,"style":250},[191],[114,15506,15508,15511,15514,15517,15520],{"className":15507},[168],[114,15509],{"className":15510,"style":15306},[172],[114,15512,5648],{"className":15513},[181,182],[114,15515],{"className":15516,"style":652},[191],[114,15518,147],{"className":15519},[656],[114,15521],{"className":15522,"style":652},[191],[114,15524,15526,15529],{"className":15525},[168],[114,15527],{"className":15528,"style":264},[172],[114,15530,756],{"className":15531,"style":771},[181,182],"，设 ",[114,15534,15536,15578],{"className":15535},[117],[114,15537,15539],{"className":15538},[121],[123,15540,15541],{"xmlns":125},[127,15542,15543,15575],{},[130,15544,15545,15551,15553,15555,15557,15559,15561,15567,15569,15571,15573],{},[8598,15546,15547,15549],{},[138,15548,1956],{},[133,15550,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[133,15552,1403],{},[138,15554,1956],{},[133,15556,147],{},[138,15558,835],{},[133,15560,144],{"separator":143},[8598,15562,15563,15565],{},[138,15564,5648],{},[133,15566,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[133,15568,1403],{},[138,15570,5648],{},[133,15572,147],{},[138,15574,11540],{},[157,15576,15577],{"encoding":159},"a' = a+x, b'=b+y",[114,15579,15581,15628,15646,15702,15720],{"className":15580,"ariaHidden":143},[164],[114,15582,15584,15587,15619,15622,15625],{"className":15583},[168],[114,15585],{"className":15586,"style":8192},[172],[114,15588,15590,15593],{"className":15589},[181],[114,15591,1956],{"className":15592},[181,182],[114,15594,15596],{"className":15595},[2010],[114,15597,15599],{"className":15598},[2014],[114,15600,15602],{"className":15601},[2019],[114,15603,15605],{"className":15604,"style":8192},[2023],[114,15606,15607,15610],{"style":8207},[114,15608],{"className":15609,"style":2032},[2031],[114,15611,15613],{"className":15612},[2036,2037,2038,2039],[114,15614,15616],{"className":15615},[181,2039],[114,15617,6991],{"className":15618},[181,2039],[114,15620],{"className":15621,"style":250},[191],[114,15623,1403],{"className":15624},[254],[114,15626],{"className":15627,"style":250},[191],[114,15629,15631,15634,15637,15640,15643],{"className":15630},[168],[114,15632],{"className":15633,"style":10327},[172],[114,15635,1956],{"className":15636},[181,182],[114,15638],{"className":15639,"style":652},[191],[114,15641,147],{"className":15642},[656],[114,15644],{"className":15645,"style":652},[191],[114,15647,15649,15652,15655,15658,15661,15693,15696,15699],{"className":15648},[168],[114,15650],{"className":15651,"style":15454},[172],[114,15653,835],{"className":15654},[181,182],[114,15656,144],{"className":15657},[187],[114,15659],{"className":15660,"style":192},[191],[114,15662,15664,15667],{"className":15663},[181],[114,15665,5648],{"className":15666},[181,182],[114,15668,15670],{"className":15669},[2010],[114,15671,15673],{"className":15672},[2014],[114,15674,15676],{"className":15675},[2019],[114,15677,15679],{"className":15678,"style":8192},[2023],[114,15680,15681,15684],{"style":8207},[114,15682],{"className":15683,"style":2032},[2031],[114,15685,15687],{"className":15686},[2036,2037,2038,2039],[114,15688,15690],{"className":15689},[181,2039],[114,15691,6991],{"className":15692},[181,2039],[114,15694],{"className":15695,"style":250},[191],[114,15697,1403],{"className":15698},[254],[114,15700],{"className":15701,"style":250},[191],[114,15703,15705,15708,15711,15714,15717],{"className":15704},[168],[114,15706],{"className":15707,"style":15306},[172],[114,15709,5648],{"className":15710},[181,182],[114,15712],{"className":15713,"style":652},[191],[114,15715,147],{"className":15716},[656],[114,15718],{"className":15719,"style":652},[191],[114,15721,15723,15726],{"className":15722},[168],[114,15724],{"className":15725,"style":2492},[172],[114,15727,11540],{"className":15728,"style":2453},[181,182]," (",[114,15731,15733,15755],{"className":15732},[117],[114,15734,15736],{"className":15735},[121],[123,15737,15738],{"xmlns":125},[127,15739,15740,15752],{},[130,15741,15742,15744,15746,15748,15750],{},[138,15743,835],{},[133,15745,144],{"separator":143},[138,15747,11540],{},[133,15749,828],{},[138,15751,756],{},[157,15753,15754],{"encoding":159},"x, y \\in I",[114,15756,15758,15785],{"className":15757,"ariaHidden":143},[164],[114,15759,15761,15764,15767,15770,15773,15776,15779,15782],{"className":15760},[168],[114,15762],{"className":15763,"style":11090},[172],[114,15765,835],{"className":15766},[181,182],[114,15768,144],{"className":15769},[187],[114,15771],{"className":15772,"style":192},[191],[114,15774,11540],{"className":15775,"style":2453},[181,182],[114,15777],{"className":15778,"style":250},[191],[114,15780,828],{"className":15781},[254],[114,15783],{"className":15784,"style":250},[191],[114,15786,15788,15791],{"className":15787},[168],[114,15789],{"className":15790,"style":264},[172],[114,15792,756],{"className":15793,"style":771},[181,182],")。\n",[114,15796,15798,15932],{"className":15797},[117],[114,15799,15801],{"className":15800},[121],[123,15802,15803],{"xmlns":125},[127,15804,15805,15929],{},[130,15806,15807,15813,15819,15821,15823,15825,15827,15829,15831,15833,15835,15837,15839,15841,15843,15845,15847,15849,15875,15877,15902,15904],{},[8598,15808,15809,15811],{},[138,15810,1956],{},[133,15812,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[8598,15814,15815,15817],{},[138,15816,5648],{},[133,15818,6991],{"mathvariant":1790,"lspace":6990,"rspace":6990},[133,15820,1403],{},[133,15822,136],{"stretchy":135},[138,15824,1956],{},[133,15826,147],{},[138,15828,835],{},[133,15830,155],{"stretchy":135},[133,15832,136],{"stretchy":135},[138,15834,5648],{},[133,15836,147],{},[138,15838,11540],{},[133,15840,155],{"stretchy":135},[133,15842,1403],{},[138,15844,1956],{},[138,15846,5648],{},[133,15848,147],{},[6950,15850,15851,15862],{},[6950,15852,15853,15859],{},[130,15854,15855,15857],{},[138,15856,1956],{},[138,15858,11540],{},[133,15860,15861],{"stretchy":143},"⏟",[130,15863,15864,15866,15868,15870,15873],{},[133,15865,828],{},[138,15867,756],{},[133,15869,136],{"stretchy":135},[6955,15871,15872],{},"左吸",[133,15874,155],{"stretchy":135},[133,15876,147],{},[6950,15878,15879,15889],{},[6950,15880,15881,15887],{},[130,15882,15883,15885],{},[138,15884,835],{},[138,15886,5648],{},[133,15888,15861],{"stretchy":143},[130,15890,15891,15893,15895,15897,15900],{},[133,15892,828],{},[138,15894,756],{},[133,15896,136],{"stretchy":135},[6955,15898,15899],{},"右吸",[133,15901,155],{"stretchy":135},[133,15903,147],{},[6950,15905,15906,15916],{},[6950,15907,15908,15914],{},[130,15909,15910,15912],{},[138,15911,835],{},[138,15913,11540],{},[133,15915,15861],{"stretchy":143},[130,15917,15918,15920,15922,15924,15927],{},[133,15919,828],{},[138,15921,756],{},[133,15923,136],{"stretchy":135},[6955,15925,15926],{},"封闭",[133,15928,155],{"stretchy":135},[157,15930,15931],{"encoding":159},"a'b' = (a+x)(b+y) = ab + \\underbrace{ay}_{\\in I (\\text{左吸})} + \\underbrace{xb}_{\\in I (\\text{右吸})} + \\underbrace{xy}_{\\in I (\\text{封闭})}",[114,15933,15935,16014,16035,16062,16083,16101,16262,16406],{"className":15934,"ariaHidden":143},[164],[114,15936,15938,15941,15973,16005,16008,16011],{"className":15937},[168],[114,15939],{"className":15940,"style":8192},[172],[114,15942,15944,15947],{"className":15943},[181],[114,15945,1956],{"className":15946},[181,182],[114,15948,15950],{"className":15949},[2010],[114,15951,15953],{"className":15952},[2014],[114,15954,15956],{"className":15955},[2019],[114,15957,15959],{"className":15958,"style":8192},[2023],[114,15960,15961,15964],{"style":8207},[114,15962],{"className":15963,"style":2032},[2031],[114,15965,15967],{"className":15966},[2036,2037,2038,2039],[114,15968,15970],{"className":15969},[181,2039],[114,15971,6991],{"className":15972},[181,2039],[114,15974,15976,15979],{"className":15975},[181],[114,15977,5648],{"className":15978},[181,182],[114,15980,15982],{"className":15981},[2010],[114,15983,15985],{"className":15984},[2014],[114,15986,15988],{"className":15987},[2019],[114,15989,15991],{"className":15990,"style":8192},[2023],[114,15992,15993,15996],{"style":8207},[114,15994],{"className":15995,"style":2032},[2031],[114,15997,15999],{"className":15998},[2036,2037,2038,2039],[114,16000,16002],{"className":16001},[181,2039],[114,16003,6991],{"className":16004},[181,2039],[114,16006],{"className":16007,"style":250},[191],[114,16009,1403],{"className":16010},[254],[114,16012],{"className":16013,"style":250},[191],[114,16015,16017,16020,16023,16026,16029,16032],{"className":16016},[168],[114,16018],{"className":16019,"style":173},[172],[114,16021,136],{"className":16022},[177],[114,16024,1956],{"className":16025},[181,182],[114,16027],{"className":16028,"style":652},[191],[114,16030,147],{"className":16031},[656],[114,16033],{"className":16034,"style":652},[191],[114,16036,16038,16041,16044,16047,16050,16053,16056,16059],{"className":16037},[168],[114,16039],{"className":16040,"style":173},[172],[114,16042,835],{"className":16043},[181,182],[114,16045,155],{"className":16046},[208],[114,16048,136],{"className":16049},[177],[114,16051,5648],{"className":16052},[181,182],[114,16054],{"className":16055,"style":652},[191],[114,16057,147],{"className":16058},[656],[114,16060],{"className":16061,"style":652},[191],[114,16063,16065,16068,16071,16074,16077,16080],{"className":16064},[168],[114,16066],{"className":16067,"style":173},[172],[114,16069,11540],{"className":16070,"style":2453},[181,182],[114,16072,155],{"className":16073},[208],[114,16075],{"className":16076,"style":250},[191],[114,16078,1403],{"className":16079},[254],[114,16081],{"className":16082,"style":250},[191],[114,16084,16086,16089,16092,16095,16098],{"className":16085},[168],[114,16087],{"className":16088,"style":15306},[172],[114,16090,13889],{"className":16091},[181,182],[114,16093],{"className":16094,"style":652},[191],[114,16096,147],{"className":16097},[656],[114,16099],{"className":16100,"style":652},[191],[114,16102,16104,16108,16253,16256,16259],{"className":16103},[168],[114,16105],{"className":16106,"style":16107},[172],"height:2.3258em;vertical-align:-1.7424em;",[114,16109,16111],{"className":16110},[7181,6950],[114,16112,16114,16244],{"className":16113},[2014,2015],[114,16115,16117,16241],{"className":16116},[2019],[114,16118,16120,16150],{"className":16119,"style":1420},[2023],[114,16121,16123,16126],{"style":16122},"top:-1.4326em;",[114,16124],{"className":16125,"style":4525},[2031],[114,16127,16129],{"className":16128},[2036,2037,2038,2039],[114,16130,16132,16135,16138,16141,16147],{"className":16131},[181,2039],[114,16133,828],{"className":16134},[254,2039],[114,16136,756],{"className":16137,"style":771},[181,182,2039],[114,16139,136],{"className":16140},[177,2039],[114,16142,16144],{"className":16143},[181,7222,2039],[114,16145,15872],{"className":16146},[181,7892,2039],[114,16148,155],{"className":16149},[208,2039],[114,16151,16153,16156],{"style":16152},"top:-3em;",[114,16154],{"className":16155,"style":4525},[2031],[114,16157,16159],{"className":16158},[7181,6950],[114,16160,16162,16232],{"className":16161},[2014,2015],[114,16163,16165,16229],{"className":16164},[2019],[114,16166,16168,16215],{"className":16167,"style":1420},[2023],[114,16169,16173,16176],{"className":16170,"style":16172},[16171],"svg-align","top:-2.1576em;",[114,16174],{"className":16175,"style":4525},[2031],[114,16177,16181,16195,16205],{"className":16178,"style":16180},[16179],"stretchy","height:0.548em;min-width:1.6em;",[114,16182,16186],{"className":16183,"style":16185},[16184],"brace-left","height:0.548em;",[7766,16187,16192],{"xmlns":7768,"width":16188,"height":16189,"viewBox":16190,"preserveAspectRatio":16191},"400em","0.548em","0 0 400000 548","xMinYMin slice",[7773,16193],{"d":16194},"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",[114,16196,16199],{"className":16197,"style":16185},[16198],"brace-center",[7766,16200,16202],{"xmlns":7768,"width":16188,"height":16189,"viewBox":16190,"preserveAspectRatio":16201},"xMidYMin slice",[7773,16203],{"d":16204},"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",[114,16206,16209],{"className":16207,"style":16185},[16208],"brace-right",[7766,16210,16212],{"xmlns":7768,"width":16188,"height":16189,"viewBox":16190,"preserveAspectRatio":16211},"xMaxYMin slice",[7773,16213],{"d":16214},"M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",[114,16216,16217,16220],{"style":16152},[114,16218],{"className":16219,"style":4525},[2031],[114,16221,16223,16226],{"className":16222},[181],[114,16224,1956],{"className":16225},[181,182],[114,16227,11540],{"className":16228,"style":2453},[181,182],[114,16230,2052],{"className":16231},[2051],[114,16233,16235],{"className":16234},[2019],[114,16236,16239],{"className":16237,"style":16238},[2023],"height:0.8424em;",[114,16240],{},[114,16242,2052],{"className":16243},[2051],[114,16245,16247],{"className":16246},[2019],[114,16248,16251],{"className":16249,"style":16250},[2023],"height:1.7424em;",[114,16252],{},[114,16254],{"className":16255,"style":652},[191],[114,16257,147],{"className":16258},[656],[114,16260],{"className":16261,"style":652},[191],[114,16263,16265,16269,16397,16400,16403],{"className":16264},[168],[114,16266],{"className":16267,"style":16268},[172],"height:2.2424em;vertical-align:-1.548em;",[114,16270,16272],{"className":16271},[7181,6950],[114,16273,16275,16388],{"className":16274},[2014,2015],[114,16276,16278,16385],{"className":16277},[2019],[114,16279,16281,16311],{"className":16280,"style":5300},[2023],[114,16282,16284,16287],{"style":16283},"top:-1.627em;",[114,16285],{"className":16286,"style":4525},[2031],[114,16288,16290],{"className":16289},[2036,2037,2038,2039],[114,16291,16293,16296,16299,16302,16308],{"className":16292},[181,2039],[114,16294,828],{"className":16295},[254,2039],[114,16297,756],{"className":16298,"style":771},[181,182,2039],[114,16300,136],{"className":16301},[177,2039],[114,16303,16305],{"className":16304},[181,7222,2039],[114,16306,15899],{"className":16307},[181,7892,2039],[114,16309,155],{"className":16310},[208,2039],[114,16312,16313,16316],{"style":16152},[114,16314],{"className":16315,"style":4525},[2031],[114,16317,16319],{"className":16318},[7181,6950],[114,16320,16322,16376],{"className":16321},[2014,2015],[114,16323,16325,16373],{"className":16324},[2019],[114,16326,16328,16359],{"className":16327,"style":5300},[2023],[114,16329,16332,16335],{"className":16330,"style":16331},[16171],"top:-2.352em;",[114,16333],{"className":16334,"style":4525},[2031],[114,16336,16338,16345,16352],{"className":16337,"style":16180},[16179],[114,16339,16341],{"className":16340,"style":16185},[16184],[7766,16342,16343],{"xmlns":7768,"width":16188,"height":16189,"viewBox":16190,"preserveAspectRatio":16191},[7773,16344],{"d":16194},[114,16346,16348],{"className":16347,"style":16185},[16198],[7766,16349,16350],{"xmlns":7768,"width":16188,"height":16189,"viewBox":16190,"preserveAspectRatio":16201},[7773,16351],{"d":16204},[114,16353,16355],{"className":16354,"style":16185},[16208],[7766,16356,16357],{"xmlns":7768,"width":16188,"height":16189,"viewBox":16190,"preserveAspectRatio":16211},[7773,16358],{"d":16214},[114,16360,16361,16364],{"style":16152},[114,16362],{"className":16363,"style":4525},[2031],[114,16365,16367,16370],{"className":16366},[181],[114,16368,835],{"className":16369},[181,182],[114,16371,5648],{"className":16372},[181,182],[114,16374,2052],{"className":16375},[2051],[114,16377,16379],{"className":16378},[2019],[114,16380,16383],{"className":16381,"style":16382},[2023],"height:0.648em;",[114,16384],{},[114,16386,2052],{"className":16387},[2051],[114,16389,16391],{"className":16390},[2019],[114,16392,16395],{"className":16393,"style":16394},[2023],"height:1.548em;",[114,16396],{},[114,16398],{"className":16399,"style":652},[191],[114,16401,147],{"className":16402},[656],[114,16404],{"className":16405,"style":652},[191],[114,16407,16409,16413],{"className":16408},[168],[114,16410],{"className":16411,"style":16412},[172],"height:2.173em;vertical-align:-1.7424em;",[114,16414,16416],{"className":16415},[7181,6950],[114,16417,16419,16529],{"className":16418},[2014,2015],[114,16420,16422,16526],{"className":16421},[2019],[114,16423,16425,16454],{"className":16424,"style":1420},[2023],[114,16426,16427,16430],{"style":16122},[114,16428],{"className":16429,"style":4525},[2031],[114,16431,16433],{"className":16432},[2036,2037,2038,2039],[114,16434,16436,16439,16442,16445,16451],{"className":16435},[181,2039],[114,16437,828],{"className":16438},[254,2039],[114,16440,756],{"className":16441,"style":771},[181,182,2039],[114,16443,136],{"className":16444},[177,2039],[114,16446,16448],{"className":16447},[181,7222,2039],[114,16449,15926],{"className":16450},[181,7892,2039],[114,16452,155],{"className":16453},[208,2039],[114,16455,16456,16459],{"style":16152},[114,16457],{"className":16458,"style":4525},[2031],[114,16460,16462],{"className":16461},[7181,6950],[114,16463,16465,16518],{"className":16464},[2014,2015],[114,16466,16468,16515],{"className":16467},[2019],[114,16469,16471,16501],{"className":16470,"style":1420},[2023],[114,16472,16474,16477],{"className":16473,"style":16172},[16171],[114,16475],{"className":16476,"style":4525},[2031],[114,16478,16480,16487,16494],{"className":16479,"style":16180},[16179],[114,16481,16483],{"className":16482,"style":16185},[16184],[7766,16484,16485],{"xmlns":7768,"width":16188,"height":16189,"viewBox":16190,"preserveAspectRatio":16191},[7773,16486],{"d":16194},[114,16488,16490],{"className":16489,"style":16185},[16198],[7766,16491,16492],{"xmlns":7768,"width":16188,"height":16189,"viewBox":16190,"preserveAspectRatio":16201},[7773,16493],{"d":16204},[114,16495,16497],{"className":16496,"style":16185},[16208],[7766,16498,16499],{"xmlns":7768,"width":16188,"height":16189,"viewBox":16190,"preserveAspectRatio":16211},[7773,16500],{"d":16214},[114,16502,16503,16506],{"style":16152},[114,16504],{"className":16505,"style":4525},[2031],[114,16507,16509,16512],{"className":16508},[181],[114,16510,835],{"className":16511},[181,182],[114,16513,11540],{"className":16514,"style":2453},[181,182],[114,16516,2052],{"className":16517},[2051],[114,16519,16521],{"className":16520},[2019],[114,16522,16524],{"className":16523,"style":16238},[2023],[114,16525],{},[114,16527,2052],{"className":16528},[2051],[114,16530,16532],{"className":16531},[2019],[114,16533,16535],{"className":16534,"style":16250},[2023],[114,16536],{},"\n差值在 ",[114,16539,16541,16554],{"className":16540},[117],[114,16542,16544],{"className":16543},[121],[123,16545,16546],{"xmlns":125},[127,16547,16548,16552],{},[130,16549,16550],{},[138,16551,756],{},[157,16553,756],{"encoding":159},[114,16555,16557],{"className":16556,"ariaHidden":143},[164],[114,16558,16560,16563],{"className":16559},[168],[114,16561],{"className":16562,"style":264},[172],[114,16564,756],{"className":16565,"style":771},[181,182]," 中，故代表元选取不影响结果。",[106,16568,16570],{"id":16569},"_62-商环的性质传承","6.2 商环的性质传承",[19,16572,16573,16574,16613,16614,16642],{},"商环 ",[114,16575,16577,16595],{"className":16576},[117],[114,16578,16580],{"className":16579},[121],[123,16581,16582],{"xmlns":125},[127,16583,16584,16592],{},[130,16585,16586,16588,16590],{},[138,16587,140],{},[138,16589,14861],{"mathvariant":1790},[138,16591,756],{},[157,16593,16594],{"encoding":159},"R\u002FI",[114,16596,16598],{"className":16597,"ariaHidden":143},[164],[114,16599,16601,16604,16607,16610],{"className":16600},[168],[114,16602],{"className":16603,"style":173},[172],[114,16605,140],{"className":16606,"style":183},[181,182],[114,16608,14861],{"className":16609},[181],[114,16611,756],{"className":16612,"style":771},[181,182]," 会继承原环 ",[114,16615,16617,16630],{"className":16616},[117],[114,16618,16620],{"className":16619},[121],[123,16621,16622],{"xmlns":125},[127,16623,16624,16628],{},[130,16625,16626],{},[138,16627,140],{},[157,16629,140],{"encoding":159},[114,16631,16633],{"className":16632,"ariaHidden":143},[164],[114,16634,16636,16639],{"className":16635},[168],[114,16637],{"className":16638,"style":264},[172],[114,16640,140],{"className":16641,"style":183},[181,182]," 的部分优良性质：",[803,16644,16645,17006],{},[450,16646,16647,455,16650,1384,16678,16716,16717],{},[51,16648,16649],{},"交换性",[114,16651,16653,16666],{"className":16652},[117],[114,16654,16656],{"className":16655},[121],[123,16657,16658],{"xmlns":125},[127,16659,16660,16664],{},[130,16661,16662],{},[138,16663,140],{},[157,16665,140],{"encoding":159},[114,16667,16669],{"className":16668,"ariaHidden":143},[164],[114,16670,16672,16675],{"className":16671},[168],[114,16673],{"className":16674,"style":264},[172],[114,16676,140],{"className":16677,"style":183},[181,182],[114,16679,16681,16698],{"className":16680},[117],[114,16682,16684],{"className":16683},[121],[123,16685,16686],{"xmlns":125},[127,16687,16688,16696],{},[130,16689,16690,16692,16694],{},[138,16691,140],{},[138,16693,14861],{"mathvariant":1790},[138,16695,756],{},[157,16697,16594],{"encoding":159},[114,16699,16701],{"className":16700,"ariaHidden":143},[164],[114,16702,16704,16707,16710,16713],{"className":16703},[168],[114,16705],{"className":16706,"style":173},[172],[114,16708,140],{"className":16709,"style":183},[181,182],[114,16711,14861],{"className":16712},[181],[114,16714,756],{"className":16715,"style":771},[181,182]," 也是交换环。\n",[447,16718,16719],{},[450,16720,16721,536,16724,445],{},[533,16722,16723],{},"理由",[114,16725,16727,16801],{"className":16726},[117],[114,16728,16730],{"className":16729},[121],[123,16731,16732],{"xmlns":125},[127,16733,16734,16798],{},[130,16735,16736,16738,16740,16742,16744,16746,16748,16750,16752,16754,16756,16758,16760,16762,16764,16766,16768,16770,16772,16774,16776,16778,16780,16782,16784,16786,16788,16790,16792,16794,16796],{},[133,16737,136],{"stretchy":135},[138,16739,1956],{},[133,16741,147],{},[138,16743,756],{},[133,16745,155],{"stretchy":135},[133,16747,136],{"stretchy":135},[138,16749,5648],{},[133,16751,147],{},[138,16753,756],{},[133,16755,155],{"stretchy":135},[133,16757,1403],{},[138,16759,1956],{},[138,16761,5648],{},[133,16763,147],{},[138,16765,756],{},[133,16767,1403],{},[138,16769,5648],{},[138,16771,1956],{},[133,16773,147],{},[138,16775,756],{},[133,16777,1403],{},[133,16779,136],{"stretchy":135},[138,16781,5648],{},[133,16783,147],{},[138,16785,756],{},[133,16787,155],{"stretchy":135},[133,16789,136],{"stretchy":135},[138,16791,1956],{},[133,16793,147],{},[138,16795,756],{},[133,16797,155],{"stretchy":135},[157,16799,16800],{"encoding":159},"(a+I)(b+I) = ab+I = ba+I = (b+I)(a+I)",[114,16802,16804,16825,16852,16873,16891,16909,16928,16946,16967,16994],{"className":16803,"ariaHidden":143},[164],[114,16805,16807,16810,16813,16816,16819,16822],{"className":16806},[168],[114,16808],{"className":16809,"style":173},[172],[114,16811,136],{"className":16812},[177],[114,16814,1956],{"className":16815},[181,182],[114,16817],{"className":16818,"style":652},[191],[114,16820,147],{"className":16821},[656],[114,16823],{"className":16824,"style":652},[191],[114,16826,16828,16831,16834,16837,16840,16843,16846,16849],{"className":16827},[168],[114,16829],{"className":16830,"style":173},[172],[114,16832,756],{"className":16833,"style":771},[181,182],[114,16835,155],{"className":16836},[208],[114,16838,136],{"className":16839},[177],[114,16841,5648],{"className":16842},[181,182],[114,16844],{"className":16845,"style":652},[191],[114,16847,147],{"className":16848},[656],[114,16850],{"className":16851,"style":652},[191],[114,16853,16855,16858,16861,16864,16867,16870],{"className":16854},[168],[114,16856],{"className":16857,"style":173},[172],[114,16859,756],{"className":16860,"style":771},[181,182],[114,16862,155],{"className":16863},[208],[114,16865],{"className":16866,"style":250},[191],[114,16868,1403],{"className":16869},[254],[114,16871],{"className":16872,"style":250},[191],[114,16874,16876,16879,16882,16885,16888],{"className":16875},[168],[114,16877],{"className":16878,"style":15306},[172],[114,16880,13889],{"className":16881},[181,182],[114,16883],{"className":16884,"style":652},[191],[114,16886,147],{"className":16887},[656],[114,16889],{"className":16890,"style":652},[191],[114,16892,16894,16897,16900,16903,16906],{"className":16893},[168],[114,16895],{"className":16896,"style":264},[172],[114,16898,756],{"className":16899,"style":771},[181,182],[114,16901],{"className":16902,"style":250},[191],[114,16904,1403],{"className":16905},[254],[114,16907],{"className":16908,"style":250},[191],[114,16910,16912,16915,16919,16922,16925],{"className":16911},[168],[114,16913],{"className":16914,"style":15306},[172],[114,16916,16918],{"className":16917},[181,182],"ba",[114,16920],{"className":16921,"style":652},[191],[114,16923,147],{"className":16924},[656],[114,16926],{"className":16927,"style":652},[191],[114,16929,16931,16934,16937,16940,16943],{"className":16930},[168],[114,16932],{"className":16933,"style":264},[172],[114,16935,756],{"className":16936,"style":771},[181,182],[114,16938],{"className":16939,"style":250},[191],[114,16941,1403],{"className":16942},[254],[114,16944],{"className":16945,"style":250},[191],[114,16947,16949,16952,16955,16958,16961,16964],{"className":16948},[168],[114,16950],{"className":16951,"style":173},[172],[114,16953,136],{"className":16954},[177],[114,16956,5648],{"className":16957},[181,182],[114,16959],{"className":16960,"style":652},[191],[114,16962,147],{"className":16963},[656],[114,16965],{"className":16966,"style":652},[191],[114,16968,16970,16973,16976,16979,16982,16985,16988,16991],{"className":16969},[168],[114,16971],{"className":16972,"style":173},[172],[114,16974,756],{"className":16975,"style":771},[181,182],[114,16977,155],{"className":16978},[208],[114,16980,136],{"className":16981},[177],[114,16983,1956],{"className":16984},[181,182],[114,16986],{"className":16987,"style":652},[191],[114,16989,147],{"className":16990},[656],[114,16992],{"className":16993,"style":652},[191],[114,16995,16997,17000,17003],{"className":16996},[168],[114,16998],{"className":16999,"style":173},[172],[114,17001,756],{"className":17002,"style":771},[181,182],[114,17004,155],{"className":17005},[208],[450,17007,17008,455,17011,17039,17040,17068,17069,17107,17108],{},[51,17009,17010],{},"单位元",[114,17012,17014,17027],{"className":17013},[117],[114,17015,17017],{"className":17016},[121],[123,17018,17019],{"xmlns":125},[127,17020,17021,17025],{},[130,17022,17023],{},[138,17024,140],{},[157,17026,140],{"encoding":159},[114,17028,17030],{"className":17029,"ariaHidden":143},[164],[114,17031,17033,17036],{"className":17032},[168],[114,17034],{"className":17035,"style":264},[172],[114,17037,140],{"className":17038,"style":183},[181,182]," 是幺环（有单位元 ",[114,17041,17043,17056],{"className":17042},[117],[114,17044,17046],{"className":17045},[121],[123,17047,17048],{"xmlns":125},[127,17049,17050,17054],{},[130,17051,17052],{},[550,17053,4373],{},[157,17055,4373],{"encoding":159},[114,17057,17059],{"className":17058,"ariaHidden":143},[164],[114,17060,17062,17065],{"className":17061},[168],[114,17063],{"className":17064,"style":2149},[172],[114,17066,4373],{"className":17067},[181],"），则 ",[114,17070,17072,17089],{"className":17071},[117],[114,17073,17075],{"className":17074},[121],[123,17076,17077],{"xmlns":125},[127,17078,17079,17087],{},[130,17080,17081,17083,17085],{},[138,17082,140],{},[138,17084,14861],{"mathvariant":1790},[138,17086,756],{},[157,17088,16594],{"encoding":159},[114,17090,17092],{"className":17091,"ariaHidden":143},[164],[114,17093,17095,17098,17101,17104],{"className":17094},[168],[114,17096],{"className":17097,"style":173},[172],[114,17099,140],{"className":17100,"style":183},[181,182],[114,17102,14861],{"className":17103},[181],[114,17105,756],{"className":17106,"style":771},[181,182]," 也是幺环。\n",[447,17109,17110,17167],{},[450,17111,17112,536,17115,445],{},[51,17113,17114],{},"单位元形式",[114,17116,17118,17136],{"className":17117},[117],[114,17119,17121],{"className":17120},[121],[123,17122,17123],{"xmlns":125},[127,17124,17125,17133],{},[130,17126,17127,17129,17131],{},[550,17128,4373],{},[133,17130,147],{},[138,17132,756],{},[157,17134,17135],{"encoding":159},"1 + I",[114,17137,17139,17158],{"className":17138,"ariaHidden":143},[164],[114,17140,17142,17146,17149,17152,17155],{"className":17141},[168],[114,17143],{"className":17144,"style":17145},[172],"height:0.7278em;vertical-align:-0.0833em;",[114,17147,4373],{"className":17148},[181],[114,17150],{"className":17151,"style":652},[191],[114,17153,147],{"className":17154},[656],[114,17156],{"className":17157,"style":652},[191],[114,17159,17161,17164],{"className":17160},[168],[114,17162],{"className":17163,"style":264},[172],[114,17165,756],{"className":17166,"style":771},[181,182],[450,17168,17169,536,17171,445],{},[533,17170,5342],{},[114,17172,17174,17226],{"className":17173},[117],[114,17175,17177],{"className":17176},[121],[123,17178,17179],{"xmlns":125},[127,17180,17181,17223],{},[130,17182,17183,17185,17187,17189,17191,17193,17195,17197,17199,17201,17203,17205,17207,17209,17211,17213,17215,17217,17219,17221],{},[133,17184,136],{"stretchy":135},[138,17186,1956],{},[133,17188,147],{},[138,17190,756],{},[133,17192,155],{"stretchy":135},[133,17194,136],{"stretchy":135},[550,17196,4373],{},[133,17198,147],{},[138,17200,756],{},[133,17202,155],{"stretchy":135},[133,17204,1403],{},[138,17206,1956],{},[133,17208,152],{},[550,17210,4373],{},[133,17212,147],{},[138,17214,756],{},[133,17216,1403],{},[138,17218,1956],{},[133,17220,147],{},[138,17222,756],{},[157,17224,17225],{"encoding":159},"(a+I)(1+I) = a \\cdot 1 + I = a + 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交集 I∩JI \\cap JI∩J",{"id":11203,"depth":17388,"text":17402},"5.2 和 I+JI + JI+J",{"id":13315,"depth":17388,"text":17404},"5.3 积 IJIJIJ",{"id":14793,"depth":17382,"text":14794,"children":17406},[17407,17408],{"id":14797,"depth":17388,"text":14798},{"id":16569,"depth":17388,"text":16570},"md",{"order":17411},4,true,"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fsubringideal",{"title":5,"description":17381},"source\u002Fmath\u002Fabstract_algebra\u002Fsubringideal","Czpsz71-cEsyO2v_jGBQJtedKwNBtMiOZR7gH-ddli8",null,1784032894520]