[{"data":1,"prerenderedAt":8623},["ShallowReactive",2],{"note:math\u002Fabstract_algebra\u002Fprimeandmaximalideal":3,"site-search-catalogue":8622},{"id":4,"title":5,"body":6,"description":8599,"extension":8614,"meta":8615,"navigation":8617,"path":8618,"seo":8619,"stem":8620,"__hash__":8621},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fprimeandmaximalideal.md","素理想与极大理想",{"type":7,"value":8,"toc":8598},"minimark",[9,14,19,27,127,130,134,139,570,812,816,937,943,1047,2242,2344,2927,2929,2933,2937,3145,3357,3420,3424,3536,3539,3644,3741,3829,3949,4041,4219,4300,4506,4695,4733,5117,5345,5445,6126,6354,6361,6570,6578,6989,7197,7199,7203,7315,7317,7321,8279,8281,8284],[10,11,13],"h1",{"id":12},"素理想与极大理想-prime-maximal-ideals","素理想与极大理想 (Prime & Maximal Ideals)",[15,16,18],"h2",{"id":17},"_0-核心逻辑图谱","0. 核心逻辑图谱",[20,21,22,23],"p",{},"核心思想：",[24,25,26],"strong",{},"理想不仅是环的子集，它还是构造新环（商环）的\"模具\"。",[28,29,30,89],"ul",{},[31,32,33,34,37,38,84,85,88],"li",{},"模具选得好（",[24,35,36],{},"素理想","） ",[39,40,43,66],"span",{"className":41},[42],"katex",[39,44,47],{"className":45},[46],"katex-mathml",[48,49,51],"math",{"xmlns":50},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[52,53,54,61],"semantics",{},[55,56,57],"mrow",{},[58,59,60],"mo",{},"⟶",[62,63,65],"annotation",{"encoding":64},"application\u002Fx-tex","\\longrightarrow",[39,67,71],{"className":68,"ariaHidden":70},[69],"katex-html","true",[39,72,75,80],{"className":73},[74],"base",[39,76],{"className":77,"style":79},[78],"strut","height:0.522em;vertical-align:-0.011em;",[39,81,60],{"className":82},[83],"mrel"," 造出的成品是 ",[24,86,87],{},"整环"," (Integral Domain)。",[31,90,91,92,37,95,84,123,126],{},"模具选得最好（",[24,93,94],{},"极大理想",[39,96,98,111],{"className":97},[42],[39,99,101],{"className":100},[46],[48,102,103],{"xmlns":50},[52,104,105,109],{},[55,106,107],{},[58,108,60],{},[62,110,65],{"encoding":64},[39,112,114],{"className":113,"ariaHidden":70},[69],[39,115,117,120],{"className":116},[74],[39,118],{"className":119,"style":79},[78],[39,121,60],{"className":122},[83],[24,124,125],{},"域"," (Field)。",[128,129],"hr",{},[15,131,133],{"id":132},"_1-素理想-prime-ideal","1. 素理想 (Prime Ideal)",[135,136,138],"h3",{"id":137},"_11-定义","1.1 定义",[20,140,141,142,176,177,207,208,305,306,376,377,432,433,485,486,537,538,566,567,569],{},"设 ",[39,143,145,160],{"className":144},[42],[39,146,148],{"className":147},[46],[48,149,150],{"xmlns":50},[52,151,152,158],{},[55,153,154],{},[155,156,157],"mi",{},"P",[62,159,157],{"encoding":64},[39,161,163],{"className":162,"ariaHidden":70},[69],[39,164,166,170],{"className":165},[74],[39,167],{"className":168,"style":169},[78],"height:0.6833em;",[39,171,157],{"className":172,"style":175},[173,174],"mord","mathnormal","margin-right:0.1389em;"," 是环 ",[39,178,180,194],{"className":179},[42],[39,181,183],{"className":182},[46],[48,184,185],{"xmlns":50},[52,186,187,192],{},[55,188,189],{},[155,190,191],{},"R",[62,193,191],{"encoding":64},[39,195,197],{"className":196,"ariaHidden":70},[69],[39,198,200,203],{"className":199},[74],[39,201],{"className":202,"style":169},[78],[39,204,191],{"className":205,"style":206},[173,174],"margin-right:0.0077em;"," 的真理想（",[39,209,211,231],{"className":210},[42],[39,212,214],{"className":213},[46],[48,215,216],{"xmlns":50},[52,217,218,228],{},[55,219,220,222,226],{},[155,221,157],{},[58,223,225],{"mathvariant":224},"normal","≠",[155,227,191],{},[62,229,230],{"encoding":64},"P \\neq R",[39,232,234,296],{"className":233,"ariaHidden":70},[69],[39,235,237,241,244,249,293],{"className":236},[74],[39,238],{"className":239,"style":240},[78],"height:0.8889em;vertical-align:-0.1944em;",[39,242,157],{"className":243,"style":175},[173,174],[39,245],{"className":246,"style":248},[247],"mspace","margin-right:0.2778em;",[39,250,252,285,289],{"className":251},[83],[39,253,255],{"className":254},[83],[39,256,259],{"className":257},[173,258],"vbox",[39,260,263],{"className":261},[262],"thinbox",[39,264,267,270,281],{"className":265},[266],"rlap",[39,268],{"className":269,"style":240},[78],[39,271,274],{"className":272},[273],"inner",[39,275,277],{"className":276},[173],[39,278,280],{"className":279},[83],"",[39,282],{"className":283},[284],"fix",[39,286],{"className":287},[247,288],"nobreak",[39,290,292],{"className":291},[83],"=",[39,294],{"className":295,"style":248},[247],[39,297,299,302],{"className":298},[74],[39,300],{"className":301,"style":169},[78],[39,303,191],{"className":304,"style":206},[173,174],"）。\n如果对于任意 ",[39,307,309,335],{"className":308},[42],[39,310,312],{"className":311},[46],[48,313,314],{"xmlns":50},[52,315,316,332],{},[55,317,318,321,324,327,330],{},[155,319,320],{},"a",[58,322,323],{"separator":70},",",[155,325,326],{},"b",[58,328,329],{},"∈",[155,331,191],{},[62,333,334],{"encoding":64},"a, b \\in R",[39,336,338,367],{"className":337,"ariaHidden":70},[69],[39,339,341,344,347,351,355,358,361,364],{"className":340},[74],[39,342],{"className":343,"style":240},[78],[39,345,320],{"className":346},[173,174],[39,348,323],{"className":349},[350],"mpunct",[39,352],{"className":353,"style":354},[247],"margin-right:0.1667em;",[39,356,326],{"className":357},[173,174],[39,359],{"className":360,"style":248},[247],[39,362,329],{"className":363},[83],[39,365],{"className":366,"style":248},[247],[39,368,370,373],{"className":369},[74],[39,371],{"className":372,"style":169},[78],[39,374,191],{"className":375,"style":206},[173,174],"，只要乘积 ",[39,378,380,400],{"className":379},[42],[39,381,383],{"className":382},[46],[48,384,385],{"xmlns":50},[52,386,387,397],{},[55,388,389,391,393,395],{},[155,390,320],{},[155,392,326],{},[58,394,329],{},[155,396,157],{},[62,398,399],{"encoding":64},"ab \\in P",[39,401,403,423],{"className":402,"ariaHidden":70},[69],[39,404,406,410,414,417,420],{"className":405},[74],[39,407],{"className":408,"style":409},[78],"height:0.7335em;vertical-align:-0.0391em;",[39,411,413],{"className":412},[173,174],"ab",[39,415],{"className":416,"style":248},[247],[39,418,329],{"className":419},[83],[39,421],{"className":422,"style":248},[247],[39,424,426,429],{"className":425},[74],[39,427],{"className":428,"style":169},[78],[39,430,157],{"className":431,"style":175},[173,174],"，就一定能推出 ",[39,434,436,454],{"className":435},[42],[39,437,439],{"className":438},[46],[48,440,441],{"xmlns":50},[52,442,443,451],{},[55,444,445,447,449],{},[155,446,320],{},[58,448,329],{},[155,450,157],{},[62,452,453],{"encoding":64},"a \\in P",[39,455,457,476],{"className":456,"ariaHidden":70},[69],[39,458,460,464,467,470,473],{"className":459},[74],[39,461],{"className":462,"style":463},[78],"height:0.5782em;vertical-align:-0.0391em;",[39,465,320],{"className":466},[173,174],[39,468],{"className":469,"style":248},[247],[39,471,329],{"className":472},[83],[39,474],{"className":475,"style":248},[247],[39,477,479,482],{"className":478},[74],[39,480],{"className":481,"style":169},[78],[39,483,157],{"className":484,"style":175},[173,174]," 或者 ",[39,487,489,507],{"className":488},[42],[39,490,492],{"className":491},[46],[48,493,494],{"xmlns":50},[52,495,496,504],{},[55,497,498,500,502],{},[155,499,326],{},[58,501,329],{},[155,503,157],{},[62,505,506],{"encoding":64},"b \\in P",[39,508,510,528],{"className":509,"ariaHidden":70},[69],[39,511,513,516,519,522,525],{"className":512},[74],[39,514],{"className":515,"style":409},[78],[39,517,326],{"className":518},[173,174],[39,520],{"className":521,"style":248},[247],[39,523,329],{"className":524},[83],[39,526],{"className":527,"style":248},[247],[39,529,531,534],{"className":530},[74],[39,532],{"className":533,"style":169},[78],[39,535,157],{"className":536,"style":175},[173,174],"。\n则称 ",[39,539,541,554],{"className":540},[42],[39,542,544],{"className":543},[46],[48,545,546],{"xmlns":50},[52,547,548,552],{},[55,549,550],{},[155,551,157],{},[62,553,157],{"encoding":64},[39,555,557],{"className":556,"ariaHidden":70},[69],[39,558,560,563],{"className":559},[74],[39,561],{"className":562,"style":169},[78],[39,564,157],{"className":565,"style":175},[173,174]," 为",[24,568,36],{},"。",[571,572,573],"blockquote",{},[20,574,575,578,579,608,609,637,638,694,695,723,724,753,754,782,783,811],{},[24,576,577],{},"直观理解","：这就像整数里的素数 ",[39,580,582,595],{"className":581},[42],[39,583,585],{"className":584},[46],[48,586,587],{"xmlns":50},[52,588,589,593],{},[55,590,591],{},[155,592,20],{},[62,594,20],{"encoding":64},[39,596,598],{"className":597,"ariaHidden":70},[69],[39,599,601,605],{"className":600},[74],[39,602],{"className":603,"style":604},[78],"height:0.625em;vertical-align:-0.1944em;",[39,606,20],{"className":607},[173,174],"。如果 ",[39,610,612,625],{"className":611},[42],[39,613,615],{"className":614},[46],[48,616,617],{"xmlns":50},[52,618,619,623],{},[55,620,621],{},[155,622,20],{},[62,624,20],{"encoding":64},[39,626,628],{"className":627,"ariaHidden":70},[69],[39,629,631,634],{"className":630},[74],[39,632],{"className":633,"style":604},[78],[39,635,20],{"className":636},[173,174]," 整除 ",[39,639,641,660],{"className":640},[42],[39,642,644],{"className":643},[46],[48,645,646],{"xmlns":50},[52,647,648,657],{},[55,649,650,652,655],{},[155,651,320],{},[58,653,654],{},"×",[155,656,326],{},[62,658,659],{"encoding":64},"a \\times b",[39,661,663,684],{"className":662,"ariaHidden":70},[69],[39,664,666,670,673,677,681],{"className":665},[74],[39,667],{"className":668,"style":669},[78],"height:0.6667em;vertical-align:-0.0833em;",[39,671,320],{"className":672},[173,174],[39,674],{"className":675,"style":676},[247],"margin-right:0.2222em;",[39,678,654],{"className":679},[680],"mbin",[39,682],{"className":683,"style":676},[247],[39,685,687,691],{"className":686},[74],[39,688],{"className":689,"style":690},[78],"height:0.6944em;",[39,692,326],{"className":693},[173,174],"，那 ",[39,696,698,711],{"className":697},[42],[39,699,701],{"className":700},[46],[48,702,703],{"xmlns":50},[52,704,705,709],{},[55,706,707],{},[155,708,20],{},[62,710,20],{"encoding":64},[39,712,714],{"className":713,"ariaHidden":70},[69],[39,715,717,720],{"className":716},[74],[39,718],{"className":719,"style":604},[78],[39,721,20],{"className":722},[173,174]," 必须整除 ",[39,725,727,740],{"className":726},[42],[39,728,730],{"className":729},[46],[48,731,732],{"xmlns":50},[52,733,734,738],{},[55,735,736],{},[155,737,320],{},[62,739,320],{"encoding":64},[39,741,743],{"className":742,"ariaHidden":70},[69],[39,744,746,750],{"className":745},[74],[39,747],{"className":748,"style":749},[78],"height:0.4306em;",[39,751,320],{"className":752},[173,174]," 或 ",[39,755,757,770],{"className":756},[42],[39,758,760],{"className":759},[46],[48,761,762],{"xmlns":50},[52,763,764,768],{},[55,765,766],{},[155,767,326],{},[62,769,326],{"encoding":64},[39,771,773],{"className":772,"ariaHidden":70},[69],[39,774,776,779],{"className":775},[74],[39,777],{"className":778,"style":690},[78],[39,780,326],{"className":781},[173,174],"。素理想 ",[39,784,786,799],{"className":785},[42],[39,787,789],{"className":788},[46],[48,790,791],{"xmlns":50},[52,792,793,797],{},[55,794,795],{},[155,796,157],{},[62,798,157],{"encoding":64},[39,800,802],{"className":801,"ariaHidden":70},[69],[39,803,805,808],{"className":804},[74],[39,806],{"className":807,"style":169},[78],[39,809,157],{"className":810,"style":175},[173,174]," 就像一个\"过滤器\"，乘积掉进去了，因子肯定至少有一个原本就在里面。",[135,813,815],{"id":814},"_12-核心定理-a-必考","1.2 核心定理 A (必考)",[20,817,818,821,822,850,851,894,895,936],{},[24,819,820],{},"定理 3.5.1","：",[39,823,825,838],{"className":824},[42],[39,826,828],{"className":827},[46],[48,829,830],{"xmlns":50},[52,831,832,836],{},[55,833,834],{},[155,835,157],{},[62,837,157],{"encoding":64},[39,839,841],{"className":840,"ariaHidden":70},[69],[39,842,844,847],{"className":843},[74],[39,845],{"className":846,"style":169},[78],[39,848,157],{"className":849,"style":175},[173,174]," 是素理想 ",[39,852,854,875],{"className":853},[42],[39,855,857],{"className":856},[46],[48,858,859],{"xmlns":50},[52,860,861,872],{},[55,862,863,867,870],{},[864,865,866],"mtext",{},"  ",[58,868,869],{},"⟺",[864,871,866],{},[62,873,874],{"encoding":64},"\\iff",[39,876,878],{"className":877,"ariaHidden":70},[69],[39,879,881,885,888,891],{"className":880},[74],[39,882],{"className":883,"style":884},[78],"height:0.549em;vertical-align:-0.024em;",[39,886],{"className":887,"style":248},[247],[39,889,869],{"className":890},[83],[39,892],{"className":893,"style":248},[247]," 商环 ",[39,896,898,917],{"className":897},[42],[39,899,901],{"className":900},[46],[48,902,903],{"xmlns":50},[52,904,905,914],{},[55,906,907,909,912],{},[155,908,191],{},[155,910,911],{"mathvariant":224},"\u002F",[155,913,157],{},[62,915,916],{"encoding":64},"R\u002FP",[39,918,920],{"className":919,"ariaHidden":70},[69],[39,921,923,927,930,933],{"className":922},[74],[39,924],{"className":925,"style":926},[78],"height:1em;vertical-align:-0.25em;",[39,928,191],{"className":929,"style":206},[173,174],[39,931,911],{"className":932},[173],[39,934,157],{"className":935,"style":175},[173,174]," 是整环。",[938,939,941],"h4",{"id":940},"证明详解",[24,942,940],{},[20,944,945],{},[24,946,947,948,979,980,1008,1009,936],{},"(",[39,949,951,966],{"className":950},[42],[39,952,954],{"className":953},[46],[48,955,956],{"xmlns":50},[52,957,958,963],{},[55,959,960],{},[58,961,962],{},"⇒",[62,964,965],{"encoding":64},"\\Rightarrow",[39,967,969],{"className":968,"ariaHidden":70},[69],[39,970,972,976],{"className":971},[74],[39,973],{"className":974,"style":975},[78],"height:0.3669em;",[39,977,962],{"className":978},[83],") 方向：已知 ",[39,981,983,996],{"className":982},[42],[39,984,986],{"className":985},[46],[48,987,988],{"xmlns":50},[52,989,990,994],{},[55,991,992],{},[155,993,157],{},[62,995,157],{"encoding":64},[39,997,999],{"className":998,"ariaHidden":70},[69],[39,1000,1002,1005],{"className":1001},[74],[39,1003],{"className":1004,"style":169},[78],[39,1006,157],{"className":1007,"style":175},[173,174]," 是素理想，求证 ",[39,1010,1012,1029],{"className":1011},[42],[39,1013,1015],{"className":1014},[46],[48,1016,1017],{"xmlns":50},[52,1018,1019,1027],{},[55,1020,1021,1023,1025],{},[155,1022,191],{},[155,1024,911],{"mathvariant":224},[155,1026,157],{},[62,1028,916],{"encoding":64},[39,1030,1032],{"className":1031,"ariaHidden":70},[69],[39,1033,1035,1038,1041,1044],{"className":1034},[74],[39,1036],{"className":1037,"style":926},[78],[39,1039,191],{"className":1040,"style":206},[173,174],[39,1042,911],{"className":1043},[173],[39,1045,157],{"className":1046,"style":175},[173,174],[1048,1049,1050,1239,1567,1981,2119,2197],"ol",{},[31,1051,1052,1055,1056,1136,1137,753,1188,569],{},[24,1053,1054],{},"回顾整环定义","：整环是交换幺环，且没有\"零因子\"。也就是说，如果 ",[39,1057,1059,1086],{"className":1058},[42],[39,1060,1062],{"className":1061},[46],[48,1063,1064],{"xmlns":50},[52,1065,1066,1083],{},[55,1067,1068,1071,1074,1077,1079],{},[155,1069,1070],{},"X",[58,1072,1073],{},"⋅",[155,1075,1076],{},"Y",[58,1078,292],{},[1080,1081,1082],"mn",{},"0",[62,1084,1085],{"encoding":64},"X \\cdot Y = 0",[39,1087,1089,1108,1126],{"className":1088,"ariaHidden":70},[69],[39,1090,1092,1095,1099,1102,1105],{"className":1091},[74],[39,1093],{"className":1094,"style":169},[78],[39,1096,1070],{"className":1097,"style":1098},[173,174],"margin-right:0.0785em;",[39,1100],{"className":1101,"style":676},[247],[39,1103,1073],{"className":1104},[680],[39,1106],{"className":1107,"style":676},[247],[39,1109,1111,1114,1117,1120,1123],{"className":1110},[74],[39,1112],{"className":1113,"style":169},[78],[39,1115,1076],{"className":1116,"style":676},[173,174],[39,1118],{"className":1119,"style":248},[247],[39,1121,292],{"className":1122},[83],[39,1124],{"className":1125,"style":248},[247],[39,1127,1129,1133],{"className":1128},[74],[39,1130],{"className":1131,"style":1132},[78],"height:0.6444em;",[39,1134,1082],{"className":1135},[173],"，则必须 ",[39,1138,1140,1158],{"className":1139},[42],[39,1141,1143],{"className":1142},[46],[48,1144,1145],{"xmlns":50},[52,1146,1147,1155],{},[55,1148,1149,1151,1153],{},[155,1150,1070],{},[58,1152,292],{},[1080,1154,1082],{},[62,1156,1157],{"encoding":64},"X=0",[39,1159,1161,1179],{"className":1160,"ariaHidden":70},[69],[39,1162,1164,1167,1170,1173,1176],{"className":1163},[74],[39,1165],{"className":1166,"style":169},[78],[39,1168,1070],{"className":1169,"style":1098},[173,174],[39,1171],{"className":1172,"style":248},[247],[39,1174,292],{"className":1175},[83],[39,1177],{"className":1178,"style":248},[247],[39,1180,1182,1185],{"className":1181},[74],[39,1183],{"className":1184,"style":1132},[78],[39,1186,1082],{"className":1187},[173],[39,1189,1191,1209],{"className":1190},[42],[39,1192,1194],{"className":1193},[46],[48,1195,1196],{"xmlns":50},[52,1197,1198,1206],{},[55,1199,1200,1202,1204],{},[155,1201,1076],{},[58,1203,292],{},[1080,1205,1082],{},[62,1207,1208],{"encoding":64},"Y=0",[39,1210,1212,1230],{"className":1211,"ariaHidden":70},[69],[39,1213,1215,1218,1221,1224,1227],{"className":1214},[74],[39,1216],{"className":1217,"style":169},[78],[39,1219,1076],{"className":1220,"style":676},[173,174],[39,1222],{"className":1223,"style":248},[247],[39,1225,292],{"className":1226},[83],[39,1228],{"className":1229,"style":248},[247],[39,1231,1233,1236],{"className":1232},[74],[39,1234],{"className":1235,"style":1132},[78],[39,1237,1082],{"className":1238},[173],[31,1240,1241,1244,1245,1283,1284,1336,1337,1389,1390],{},[24,1242,1243],{},"设定场景","：在商环 ",[39,1246,1248,1265],{"className":1247},[42],[39,1249,1251],{"className":1250},[46],[48,1252,1253],{"xmlns":50},[52,1254,1255,1263],{},[55,1256,1257,1259,1261],{},[155,1258,191],{},[155,1260,911],{"mathvariant":224},[155,1262,157],{},[62,1264,916],{"encoding":64},[39,1266,1268],{"className":1267,"ariaHidden":70},[69],[39,1269,1271,1274,1277,1280],{"className":1270},[74],[39,1272],{"className":1273,"style":926},[78],[39,1275,191],{"className":1276,"style":206},[173,174],[39,1278,911],{"className":1279},[173],[39,1281,157],{"className":1282,"style":175},[173,174]," 中，取两个非零元素 ",[39,1285,1287,1306],{"className":1286},[42],[39,1288,1290],{"className":1289},[46],[48,1291,1292],{"xmlns":50},[52,1293,1294,1303],{},[55,1295,1296,1298,1301],{},[155,1297,320],{},[58,1299,1300],{},"+",[155,1302,157],{},[62,1304,1305],{"encoding":64},"a+P",[39,1307,1309,1327],{"className":1308,"ariaHidden":70},[69],[39,1310,1312,1315,1318,1321,1324],{"className":1311},[74],[39,1313],{"className":1314,"style":669},[78],[39,1316,320],{"className":1317},[173,174],[39,1319],{"className":1320,"style":676},[247],[39,1322,1300],{"className":1323},[680],[39,1325],{"className":1326,"style":676},[247],[39,1328,1330,1333],{"className":1329},[74],[39,1331],{"className":1332,"style":169},[78],[39,1334,157],{"className":1335,"style":175},[173,174]," 和 ",[39,1338,1340,1358],{"className":1339},[42],[39,1341,1343],{"className":1342},[46],[48,1344,1345],{"xmlns":50},[52,1346,1347,1355],{},[55,1348,1349,1351,1353],{},[155,1350,326],{},[58,1352,1300],{},[155,1354,157],{},[62,1356,1357],{"encoding":64},"b+P",[39,1359,1361,1380],{"className":1360,"ariaHidden":70},[69],[39,1362,1364,1368,1371,1374,1377],{"className":1363},[74],[39,1365],{"className":1366,"style":1367},[78],"height:0.7778em;vertical-align:-0.0833em;",[39,1369,326],{"className":1370},[173,174],[39,1372],{"className":1373,"style":676},[247],[39,1375,1300],{"className":1376},[680],[39,1378],{"className":1379,"style":676},[247],[39,1381,1383,1386],{"className":1382},[74],[39,1384],{"className":1385,"style":169},[78],[39,1387,157],{"className":1388,"style":175},[173,174],"。\n",[28,1391,1392],{},[31,1393,1394,1395,1482,1483,569],{},"这里的\"非零\"意味着 ",[39,1396,1398,1417],{"className":1397},[42],[39,1399,1401],{"className":1400},[46],[48,1402,1403],{"xmlns":50},[52,1404,1405,1414],{},[55,1406,1407,1409,1412],{},[155,1408,320],{},[58,1410,1411],{"mathvariant":224},"∉",[155,1413,157],{},[62,1415,1416],{"encoding":64},"a \\notin P",[39,1418,1420,1473],{"className":1419,"ariaHidden":70},[69],[39,1421,1423,1426,1429,1432,1470],{"className":1422},[74],[39,1424],{"className":1425,"style":926},[78],[39,1427,320],{"className":1428},[173,174],[39,1430],{"className":1431,"style":248},[247],[39,1433,1435,1441],{"className":1434},[83],[39,1436,1438],{"className":1437},[173],[39,1439,329],{"className":1440},[83],[39,1442,1444],{"className":1443},[173,258],[39,1445,1447],{"className":1446},[262],[39,1448,1451,1454,1467],{"className":1449},[1450],"llap",[39,1452],{"className":1453,"style":926},[78],[39,1455,1457],{"className":1456},[273],[39,1458,1460,1463],{"className":1459},[173],[39,1461,911],{"className":1462},[173],[39,1464],{"className":1465,"style":1466},[247],"margin-right:0.0556em;",[39,1468],{"className":1469},[284],[39,1471],{"className":1472,"style":248},[247],[39,1474,1476,1479],{"className":1475},[74],[39,1477],{"className":1478,"style":169},[78],[39,1480,157],{"className":1481,"style":175},[173,174]," 且 ",[39,1484,1486,1504],{"className":1485},[42],[39,1487,1489],{"className":1488},[46],[48,1490,1491],{"xmlns":50},[52,1492,1493,1501],{},[55,1494,1495,1497,1499],{},[155,1496,326],{},[58,1498,1411],{"mathvariant":224},[155,1500,157],{},[62,1502,1503],{"encoding":64},"b \\notin P",[39,1505,1507,1558],{"className":1506,"ariaHidden":70},[69],[39,1508,1510,1513,1516,1519,1555],{"className":1509},[74],[39,1511],{"className":1512,"style":926},[78],[39,1514,326],{"className":1515},[173,174],[39,1517],{"className":1518,"style":248},[247],[39,1520,1522,1528],{"className":1521},[83],[39,1523,1525],{"className":1524},[173],[39,1526,329],{"className":1527},[83],[39,1529,1531],{"className":1530},[173,258],[39,1532,1534],{"className":1533},[262],[39,1535,1537,1540,1552],{"className":1536},[1450],[39,1538],{"className":1539,"style":926},[78],[39,1541,1543],{"className":1542},[273],[39,1544,1546,1549],{"className":1545},[173],[39,1547,911],{"className":1548},[173],[39,1550],{"className":1551,"style":1466},[247],[39,1553],{"className":1554},[284],[39,1556],{"className":1557,"style":248},[247],[39,1559,1561,1564],{"className":1560},[74],[39,1562],{"className":1563,"style":169},[78],[39,1565,157],{"className":1566,"style":175},[173,174],[31,1568,1569,1572,1573,1389,1771],{},[24,1570,1571],{},"假设乘积为零","：假设 ",[39,1574,1576,1625],{"className":1575},[42],[39,1577,1579],{"className":1578},[46],[48,1580,1581],{"xmlns":50},[52,1582,1583,1622],{},[55,1584,1585,1588,1590,1592,1594,1597,1599,1601,1603,1605,1607,1609],{},[58,1586,947],{"stretchy":1587},"false",[155,1589,320],{},[58,1591,1300],{},[155,1593,157],{},[58,1595,1596],{"stretchy":1587},")",[58,1598,947],{"stretchy":1587},[155,1600,326],{},[58,1602,1300],{},[155,1604,157],{},[58,1606,1596],{"stretchy":1587},[58,1608,292],{},[1610,1611,1612,1614],"msub",{},[1080,1613,1082],{},[55,1615,1616,1618,1620],{},[155,1617,191],{},[155,1619,911],{"mathvariant":224},[155,1621,157],{},[62,1623,1624],{"encoding":64},"(a+P)(b+P) = 0_{R\u002FP}",[39,1626,1628,1650,1678,1699],{"className":1627,"ariaHidden":70},[69],[39,1629,1631,1634,1638,1641,1644,1647],{"className":1630},[74],[39,1632],{"className":1633,"style":926},[78],[39,1635,947],{"className":1636},[1637],"mopen",[39,1639,320],{"className":1640},[173,174],[39,1642],{"className":1643,"style":676},[247],[39,1645,1300],{"className":1646},[680],[39,1648],{"className":1649,"style":676},[247],[39,1651,1653,1656,1659,1663,1666,1669,1672,1675],{"className":1652},[74],[39,1654],{"className":1655,"style":926},[78],[39,1657,157],{"className":1658,"style":175},[173,174],[39,1660,1596],{"className":1661},[1662],"mclose",[39,1664,947],{"className":1665},[1637],[39,1667,326],{"className":1668},[173,174],[39,1670],{"className":1671,"style":676},[247],[39,1673,1300],{"className":1674},[680],[39,1676],{"className":1677,"style":676},[247],[39,1679,1681,1684,1687,1690,1693,1696],{"className":1680},[74],[39,1682],{"className":1683,"style":926},[78],[39,1685,157],{"className":1686,"style":175},[173,174],[39,1688,1596],{"className":1689},[1662],[39,1691],{"className":1692,"style":248},[247],[39,1694,292],{"className":1695},[83],[39,1697],{"className":1698,"style":248},[247],[39,1700,1702,1706],{"className":1701},[74],[39,1703],{"className":1704,"style":1705},[78],"height:0.9996em;vertical-align:-0.3552em;",[39,1707,1709,1712],{"className":1708},[173],[39,1710,1082],{"className":1711},[173],[39,1713,1716],{"className":1714},[1715],"msupsub",[39,1717,1721,1762],{"className":1718},[1719,1720],"vlist-t","vlist-t2",[39,1722,1725,1757],{"className":1723},[1724],"vlist-r",[39,1726,1730],{"className":1727,"style":1729},[1728],"vlist","height:0.3448em;",[39,1731,1733,1738],{"style":1732},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[39,1734],{"className":1735,"style":1737},[1736],"pstrut","height:2.7em;",[39,1739,1745],{"className":1740},[1741,1742,1743,1744],"sizing","reset-size6","size3","mtight",[39,1746,1748,1751,1754],{"className":1747},[173,1744],[39,1749,191],{"className":1750,"style":206},[173,174,1744],[39,1752,911],{"className":1753},[173,1744],[39,1755,157],{"className":1756,"style":175},[173,174,1744],[39,1758,1761],{"className":1759},[1760],"vlist-s","​",[39,1763,1765],{"className":1764},[1724],[39,1766,1769],{"className":1767,"style":1768},[1728],"height:0.3552em;",[39,1770],{},[28,1772,1773,1850],{},[31,1774,1775,1776,569],{},"商环的零元是 ",[39,1777,1779,1801],{"className":1778},[42],[39,1780,1782],{"className":1781},[46],[48,1783,1784],{"xmlns":50},[52,1785,1786,1798],{},[55,1787,1788,1790,1792,1794,1796],{},[1080,1789,1082],{},[58,1791,1300],{},[155,1793,157],{},[58,1795,292],{},[155,1797,157],{},[62,1799,1800],{"encoding":64},"0+P = P",[39,1802,1804,1823,1841],{"className":1803,"ariaHidden":70},[69],[39,1805,1807,1811,1814,1817,1820],{"className":1806},[74],[39,1808],{"className":1809,"style":1810},[78],"height:0.7278em;vertical-align:-0.0833em;",[39,1812,1082],{"className":1813},[173],[39,1815],{"className":1816,"style":676},[247],[39,1818,1300],{"className":1819},[680],[39,1821],{"className":1822,"style":676},[247],[39,1824,1826,1829,1832,1835,1838],{"className":1825},[74],[39,1827],{"className":1828,"style":169},[78],[39,1830,157],{"className":1831,"style":175},[173,174],[39,1833],{"className":1834,"style":248},[247],[39,1836,292],{"className":1837},[83],[39,1839],{"className":1840,"style":248},[247],[39,1842,1844,1847],{"className":1843},[74],[39,1845],{"className":1846,"style":169},[78],[39,1848,157],{"className":1849,"style":175},[173,174],[31,1851,1852,1853,1928,1929,569],{},"这意味着 ",[39,1854,1856,1880],{"className":1855},[42],[39,1857,1859],{"className":1858},[46],[48,1860,1861],{"xmlns":50},[52,1862,1863,1877],{},[55,1864,1865,1867,1869,1871,1873,1875],{},[155,1866,320],{},[155,1868,326],{},[58,1870,1300],{},[155,1872,157],{},[58,1874,292],{},[155,1876,157],{},[62,1878,1879],{"encoding":64},"ab + P = P",[39,1881,1883,1901,1919],{"className":1882,"ariaHidden":70},[69],[39,1884,1886,1889,1892,1895,1898],{"className":1885},[74],[39,1887],{"className":1888,"style":1367},[78],[39,1890,413],{"className":1891},[173,174],[39,1893],{"className":1894,"style":676},[247],[39,1896,1300],{"className":1897},[680],[39,1899],{"className":1900,"style":676},[247],[39,1902,1904,1907,1910,1913,1916],{"className":1903},[74],[39,1905],{"className":1906,"style":169},[78],[39,1908,157],{"className":1909,"style":175},[173,174],[39,1911],{"className":1912,"style":248},[247],[39,1914,292],{"className":1915},[83],[39,1917],{"className":1918,"style":248},[247],[39,1920,1922,1925],{"className":1921},[74],[39,1923],{"className":1924,"style":169},[78],[39,1926,157],{"className":1927,"style":175},[173,174],"，即 ",[39,1930,1932,1951],{"className":1931},[42],[39,1933,1935],{"className":1934},[46],[48,1936,1937],{"xmlns":50},[52,1938,1939,1949],{},[55,1940,1941,1943,1945,1947],{},[155,1942,320],{},[155,1944,326],{},[58,1946,329],{},[155,1948,157],{},[62,1950,399],{"encoding":64},[39,1952,1954,1972],{"className":1953,"ariaHidden":70},[69],[39,1955,1957,1960,1963,1966,1969],{"className":1956},[74],[39,1958],{"className":1959,"style":409},[78],[39,1961,413],{"className":1962},[173,174],[39,1964],{"className":1965,"style":248},[247],[39,1967,329],{"className":1968},[83],[39,1970],{"className":1971,"style":248},[247],[39,1973,1975,1978],{"className":1974},[74],[39,1976],{"className":1977,"style":169},[78],[39,1979,157],{"className":1980,"style":175},[173,174],[31,1982,1983,1986,1987,2015,2016,2018,2019,485,2069,569],{},[24,1984,1985],{},"利用已知条件","：因为 ",[39,1988,1990,2003],{"className":1989},[42],[39,1991,1993],{"className":1992},[46],[48,1994,1995],{"xmlns":50},[52,1996,1997,2001],{},[55,1998,1999],{},[155,2000,157],{},[62,2002,157],{"encoding":64},[39,2004,2006],{"className":2005,"ariaHidden":70},[69],[39,2007,2009,2012],{"className":2008},[74],[39,2010],{"className":2011,"style":169},[78],[39,2013,157],{"className":2014,"style":175},[173,174]," 是",[24,2017,36],{},"，由定义可知：",[39,2020,2022,2039],{"className":2021},[42],[39,2023,2025],{"className":2024},[46],[48,2026,2027],{"xmlns":50},[52,2028,2029,2037],{},[55,2030,2031,2033,2035],{},[155,2032,320],{},[58,2034,329],{},[155,2036,157],{},[62,2038,453],{"encoding":64},[39,2040,2042,2060],{"className":2041,"ariaHidden":70},[69],[39,2043,2045,2048,2051,2054,2057],{"className":2044},[74],[39,2046],{"className":2047,"style":463},[78],[39,2049,320],{"className":2050},[173,174],[39,2052],{"className":2053,"style":248},[247],[39,2055,329],{"className":2056},[83],[39,2058],{"className":2059,"style":248},[247],[39,2061,2063,2066],{"className":2062},[74],[39,2064],{"className":2065,"style":169},[78],[39,2067,157],{"className":2068,"style":175},[173,174],[39,2070,2072,2089],{"className":2071},[42],[39,2073,2075],{"className":2074},[46],[48,2076,2077],{"xmlns":50},[52,2078,2079,2087],{},[55,2080,2081,2083,2085],{},[155,2082,326],{},[58,2084,329],{},[155,2086,157],{},[62,2088,506],{"encoding":64},[39,2090,2092,2110],{"className":2091,"ariaHidden":70},[69],[39,2093,2095,2098,2101,2104,2107],{"className":2094},[74],[39,2096],{"className":2097,"style":409},[78],[39,2099,326],{"className":2100},[173,174],[39,2102],{"className":2103,"style":248},[247],[39,2105,329],{"className":2106},[83],[39,2108],{"className":2109,"style":248},[247],[39,2111,2113,2116],{"className":2112},[74],[39,2114],{"className":2115,"style":169},[78],[39,2117,157],{"className":2118,"style":175},[173,174],[31,2120,2121,2124,2125,2167,2168,2196],{},[24,2122,2123],{},"矛盾\u002F结论","：但这与步骤2中\"",[39,2126,2128,2146],{"className":2127},[42],[39,2129,2131],{"className":2130},[46],[48,2132,2133],{"xmlns":50},[52,2134,2135,2143],{},[55,2136,2137,2139,2141],{},[155,2138,320],{},[58,2140,323],{"separator":70},[155,2142,326],{},[62,2144,2145],{"encoding":64},"a, b",[39,2147,2149],{"className":2148,"ariaHidden":70},[69],[39,2150,2152,2155,2158,2161,2164],{"className":2151},[74],[39,2153],{"className":2154,"style":240},[78],[39,2156,320],{"className":2157},[173,174],[39,2159,323],{"className":2160},[350],[39,2162],{"className":2163,"style":354},[247],[39,2165,326],{"className":2166},[173,174]," 都不在 ",[39,2169,2171,2184],{"className":2170},[42],[39,2172,2174],{"className":2173},[46],[48,2175,2176],{"xmlns":50},[52,2177,2178,2182],{},[55,2179,2180],{},[155,2181,157],{},[62,2183,157],{"encoding":64},[39,2185,2187],{"className":2186,"ariaHidden":70},[69],[39,2188,2190,2193],{"className":2189},[74],[39,2191],{"className":2192,"style":169},[78],[39,2194,157],{"className":2195,"style":175},[173,174]," 中\"矛盾。如果乘积是0，因子必有一个是0。",[31,2198,2199,2202,2203,2241],{},[24,2200,2201],{},"证毕","：所以 ",[39,2204,2206,2223],{"className":2205},[42],[39,2207,2209],{"className":2208},[46],[48,2210,2211],{"xmlns":50},[52,2212,2213,2221],{},[55,2214,2215,2217,2219],{},[155,2216,191],{},[155,2218,911],{"mathvariant":224},[155,2220,157],{},[62,2222,916],{"encoding":64},[39,2224,2226],{"className":2225,"ariaHidden":70},[69],[39,2227,2229,2232,2235,2238],{"className":2228},[74],[39,2230],{"className":2231,"style":926},[78],[39,2233,191],{"className":2234,"style":206},[173,174],[39,2236,911],{"className":2237},[173],[39,2239,157],{"className":2240,"style":175},[173,174]," 没有零因子，是整环。",[20,2243,2244],{},[24,2245,947,2246,979,2276,2314,2315,2343],{},[39,2247,2249,2264],{"className":2248},[42],[39,2250,2252],{"className":2251},[46],[48,2253,2254],{"xmlns":50},[52,2255,2256,2261],{},[55,2257,2258],{},[58,2259,2260],{},"⇐",[62,2262,2263],{"encoding":64},"\\Leftarrow",[39,2265,2267],{"className":2266,"ariaHidden":70},[69],[39,2268,2270,2273],{"className":2269},[74],[39,2271],{"className":2272,"style":975},[78],[39,2274,2260],{"className":2275},[83],[39,2277,2279,2296],{"className":2278},[42],[39,2280,2282],{"className":2281},[46],[48,2283,2284],{"xmlns":50},[52,2285,2286,2294],{},[55,2287,2288,2290,2292],{},[155,2289,191],{},[155,2291,911],{"mathvariant":224},[155,2293,157],{},[62,2295,916],{"encoding":64},[39,2297,2299],{"className":2298,"ariaHidden":70},[69],[39,2300,2302,2305,2308,2311],{"className":2301},[74],[39,2303],{"className":2304,"style":926},[78],[39,2306,191],{"className":2307,"style":206},[173,174],[39,2309,911],{"className":2310},[173],[39,2312,157],{"className":2313,"style":175},[173,174]," 是整环，求证 ",[39,2316,2318,2331],{"className":2317},[42],[39,2319,2321],{"className":2320},[46],[48,2322,2323],{"xmlns":50},[52,2324,2325,2329],{},[55,2326,2327],{},[155,2328,157],{},[62,2330,157],{"encoding":64},[39,2332,2334],{"className":2333,"ariaHidden":70},[69],[39,2335,2337,2340],{"className":2336},[74],[39,2338],{"className":2339,"style":169},[78],[39,2341,157],{"className":2342,"style":175},[173,174]," 是素理想。",[1048,2345,2346,2402,2625,2816,2922],{},[31,2347,2348,1572,2350,569],{},[24,2349,1243],{},[39,2351,2353,2372],{"className":2352},[42],[39,2354,2356],{"className":2355},[46],[48,2357,2358],{"xmlns":50},[52,2359,2360,2370],{},[55,2361,2362,2364,2366,2368],{},[155,2363,320],{},[155,2365,326],{},[58,2367,329],{},[155,2369,157],{},[62,2371,399],{"encoding":64},[39,2373,2375,2393],{"className":2374,"ariaHidden":70},[69],[39,2376,2378,2381,2384,2387,2390],{"className":2377},[74],[39,2379],{"className":2380,"style":409},[78],[39,2382,413],{"className":2383},[173,174],[39,2385],{"className":2386,"style":248},[247],[39,2388,329],{"className":2389},[83],[39,2391],{"className":2392,"style":248},[247],[39,2394,2396,2399],{"className":2395},[74],[39,2397],{"className":2398,"style":169},[78],[39,2400,157],{"className":2401,"style":175},[173,174],[31,2403,2404,2407,2408,1928,2505,569],{},[24,2405,2406],{},"翻译到商环","：这意味着在商环里，",[39,2409,2411,2439],{"className":2410},[42],[39,2412,2414],{"className":2413},[46],[48,2415,2416],{"xmlns":50},[52,2417,2418,2436],{},[55,2419,2420,2422,2424,2426,2428,2430,2432,2434],{},[155,2421,320],{},[155,2423,326],{},[58,2425,1300],{},[155,2427,157],{},[58,2429,292],{},[1080,2431,1082],{},[58,2433,1300],{},[155,2435,157],{},[62,2437,2438],{"encoding":64},"ab+P = 0+P",[39,2440,2442,2460,2478,2496],{"className":2441,"ariaHidden":70},[69],[39,2443,2445,2448,2451,2454,2457],{"className":2444},[74],[39,2446],{"className":2447,"style":1367},[78],[39,2449,413],{"className":2450},[173,174],[39,2452],{"className":2453,"style":676},[247],[39,2455,1300],{"className":2456},[680],[39,2458],{"className":2459,"style":676},[247],[39,2461,2463,2466,2469,2472,2475],{"className":2462},[74],[39,2464],{"className":2465,"style":169},[78],[39,2467,157],{"className":2468,"style":175},[173,174],[39,2470],{"className":2471,"style":248},[247],[39,2473,292],{"className":2474},[83],[39,2476],{"className":2477,"style":248},[247],[39,2479,2481,2484,2487,2490,2493],{"className":2480},[74],[39,2482],{"className":2483,"style":1810},[78],[39,2485,1082],{"className":2486},[173],[39,2488],{"className":2489,"style":676},[247],[39,2491,1300],{"className":2492},[680],[39,2494],{"className":2495,"style":676},[247],[39,2497,2499,2502],{"className":2498},[74],[39,2500],{"className":2501,"style":169},[78],[39,2503,157],{"className":2504,"style":175},[173,174],[39,2506,2508,2544],{"className":2507},[42],[39,2509,2511],{"className":2510},[46],[48,2512,2513],{"xmlns":50},[52,2514,2515,2541],{},[55,2516,2517,2519,2521,2523,2525,2527,2529,2531,2533,2535,2537,2539],{},[58,2518,947],{"stretchy":1587},[155,2520,320],{},[58,2522,1300],{},[155,2524,157],{},[58,2526,1596],{"stretchy":1587},[58,2528,947],{"stretchy":1587},[155,2530,326],{},[58,2532,1300],{},[155,2534,157],{},[58,2536,1596],{"stretchy":1587},[58,2538,292],{},[1080,2540,1082],{},[62,2542,2543],{"encoding":64},"(a+P)(b+P) = 0",[39,2545,2547,2568,2595,2616],{"className":2546,"ariaHidden":70},[69],[39,2548,2550,2553,2556,2559,2562,2565],{"className":2549},[74],[39,2551],{"className":2552,"style":926},[78],[39,2554,947],{"className":2555},[1637],[39,2557,320],{"className":2558},[173,174],[39,2560],{"className":2561,"style":676},[247],[39,2563,1300],{"className":2564},[680],[39,2566],{"className":2567,"style":676},[247],[39,2569,2571,2574,2577,2580,2583,2586,2589,2592],{"className":2570},[74],[39,2572],{"className":2573,"style":926},[78],[39,2575,157],{"className":2576,"style":175},[173,174],[39,2578,1596],{"className":2579},[1662],[39,2581,947],{"className":2582},[1637],[39,2584,326],{"className":2585},[173,174],[39,2587],{"className":2588,"style":676},[247],[39,2590,1300],{"className":2591},[680],[39,2593],{"className":2594,"style":676},[247],[39,2596,2598,2601,2604,2607,2610,2613],{"className":2597},[74],[39,2599],{"className":2600,"style":926},[78],[39,2602,157],{"className":2603,"style":175},[173,174],[39,2605,1596],{"className":2606},[1662],[39,2608],{"className":2609,"style":248},[247],[39,2611,292],{"className":2612},[83],[39,2614],{"className":2615,"style":248},[247],[39,2617,2619,2622],{"className":2618},[74],[39,2620],{"className":2621,"style":1132},[78],[39,2623,1082],{"className":2624},[173],[31,2626,2627,1986,2629,2015,2667,2669,2670,485,2743,569],{},[24,2628,1985],{},[39,2630,2632,2649],{"className":2631},[42],[39,2633,2635],{"className":2634},[46],[48,2636,2637],{"xmlns":50},[52,2638,2639,2647],{},[55,2640,2641,2643,2645],{},[155,2642,191],{},[155,2644,911],{"mathvariant":224},[155,2646,157],{},[62,2648,916],{"encoding":64},[39,2650,2652],{"className":2651,"ariaHidden":70},[69],[39,2653,2655,2658,2661,2664],{"className":2654},[74],[39,2656],{"className":2657,"style":926},[78],[39,2659,191],{"className":2660,"style":206},[173,174],[39,2662,911],{"className":2663},[173],[39,2665,157],{"className":2666,"style":175},[173,174],[24,2668,87],{},"，没有零因子，所以必须有 ",[39,2671,2673,2695],{"className":2672},[42],[39,2674,2676],{"className":2675},[46],[48,2677,2678],{"xmlns":50},[52,2679,2680,2692],{},[55,2681,2682,2684,2686,2688,2690],{},[155,2683,320],{},[58,2685,1300],{},[155,2687,157],{},[58,2689,292],{},[1080,2691,1082],{},[62,2693,2694],{"encoding":64},"a+P=0",[39,2696,2698,2716,2734],{"className":2697,"ariaHidden":70},[69],[39,2699,2701,2704,2707,2710,2713],{"className":2700},[74],[39,2702],{"className":2703,"style":669},[78],[39,2705,320],{"className":2706},[173,174],[39,2708],{"className":2709,"style":676},[247],[39,2711,1300],{"className":2712},[680],[39,2714],{"className":2715,"style":676},[247],[39,2717,2719,2722,2725,2728,2731],{"className":2718},[74],[39,2720],{"className":2721,"style":169},[78],[39,2723,157],{"className":2724,"style":175},[173,174],[39,2726],{"className":2727,"style":248},[247],[39,2729,292],{"className":2730},[83],[39,2732],{"className":2733,"style":248},[247],[39,2735,2737,2740],{"className":2736},[74],[39,2738],{"className":2739,"style":1132},[78],[39,2741,1082],{"className":2742},[173],[39,2744,2746,2768],{"className":2745},[42],[39,2747,2749],{"className":2748},[46],[48,2750,2751],{"xmlns":50},[52,2752,2753,2765],{},[55,2754,2755,2757,2759,2761,2763],{},[155,2756,326],{},[58,2758,1300],{},[155,2760,157],{},[58,2762,292],{},[1080,2764,1082],{},[62,2766,2767],{"encoding":64},"b+P=0",[39,2769,2771,2789,2807],{"className":2770,"ariaHidden":70},[69],[39,2772,2774,2777,2780,2783,2786],{"className":2773},[74],[39,2775],{"className":2776,"style":1367},[78],[39,2778,326],{"className":2779},[173,174],[39,2781],{"className":2782,"style":676},[247],[39,2784,1300],{"className":2785},[680],[39,2787],{"className":2788,"style":676},[247],[39,2790,2792,2795,2798,2801,2804],{"className":2791},[74],[39,2793],{"className":2794,"style":169},[78],[39,2796,157],{"className":2797,"style":175},[173,174],[39,2799],{"className":2800,"style":248},[247],[39,2802,292],{"className":2803},[83],[39,2805],{"className":2806,"style":248},[247],[39,2808,2810,2813],{"className":2809},[74],[39,2811],{"className":2812,"style":1132},[78],[39,2814,1082],{"className":2815},[173],[31,2817,2818,2821,2822,485,2872,569],{},[24,2819,2820],{},"还原","：即 ",[39,2823,2825,2842],{"className":2824},[42],[39,2826,2828],{"className":2827},[46],[48,2829,2830],{"xmlns":50},[52,2831,2832,2840],{},[55,2833,2834,2836,2838],{},[155,2835,320],{},[58,2837,329],{},[155,2839,157],{},[62,2841,453],{"encoding":64},[39,2843,2845,2863],{"className":2844,"ariaHidden":70},[69],[39,2846,2848,2851,2854,2857,2860],{"className":2847},[74],[39,2849],{"className":2850,"style":463},[78],[39,2852,320],{"className":2853},[173,174],[39,2855],{"className":2856,"style":248},[247],[39,2858,329],{"className":2859},[83],[39,2861],{"className":2862,"style":248},[247],[39,2864,2866,2869],{"className":2865},[74],[39,2867],{"className":2868,"style":169},[78],[39,2870,157],{"className":2871,"style":175},[173,174],[39,2873,2875,2892],{"className":2874},[42],[39,2876,2878],{"className":2877},[46],[48,2879,2880],{"xmlns":50},[52,2881,2882,2890],{},[55,2883,2884,2886,2888],{},[155,2885,326],{},[58,2887,329],{},[155,2889,157],{},[62,2891,506],{"encoding":64},[39,2893,2895,2913],{"className":2894,"ariaHidden":70},[69],[39,2896,2898,2901,2904,2907,2910],{"className":2897},[74],[39,2899],{"className":2900,"style":409},[78],[39,2902,326],{"className":2903},[173,174],[39,2905],{"className":2906,"style":248},[247],[39,2908,329],{"className":2909},[83],[39,2911],{"className":2912,"style":248},[247],[39,2914,2916,2919],{"className":2915},[74],[39,2917],{"className":2918,"style":169},[78],[39,2920,157],{"className":2921,"style":175},[173,174],[31,2923,2924,2926],{},[24,2925,2201],{},"：符合素理想定义。",[128,2928],{},[15,2930,2932],{"id":2931},"_2-极大理想-maximal-ideal","2. 极大理想 (Maximal Ideal)",[135,2934,2936],{"id":2935},"_21-定义","2.1 定义",[20,2938,141,2939,176,2969,2997,2998,3058,3059,1336,3087,3115,3116,3144],{},[39,2940,2942,2956],{"className":2941},[42],[39,2943,2945],{"className":2944},[46],[48,2946,2947],{"xmlns":50},[52,2948,2949,2954],{},[55,2950,2951],{},[155,2952,2953],{},"M",[62,2955,2953],{"encoding":64},[39,2957,2959],{"className":2958,"ariaHidden":70},[69],[39,2960,2962,2965],{"className":2961},[74],[39,2963],{"className":2964,"style":169},[78],[39,2966,2953],{"className":2967,"style":2968},[173,174],"margin-right:0.109em;",[39,2970,2972,2985],{"className":2971},[42],[39,2973,2975],{"className":2974},[46],[48,2976,2977],{"xmlns":50},[52,2978,2979,2983],{},[55,2980,2981],{},[155,2982,191],{},[62,2984,191],{"encoding":64},[39,2986,2988],{"className":2987,"ariaHidden":70},[69],[39,2989,2991,2994],{"className":2990},[74],[39,2992],{"className":2993,"style":169},[78],[39,2995,191],{"className":2996,"style":206},[173,174]," 的真理想。如果",[24,2999,3000,3001,1336,3029,3057],{},"在 ",[39,3002,3004,3017],{"className":3003},[42],[39,3005,3007],{"className":3006},[46],[48,3008,3009],{"xmlns":50},[52,3010,3011,3015],{},[55,3012,3013],{},[155,3014,2953],{},[62,3016,2953],{"encoding":64},[39,3018,3020],{"className":3019,"ariaHidden":70},[69],[39,3021,3023,3026],{"className":3022},[74],[39,3024],{"className":3025,"style":169},[78],[39,3027,2953],{"className":3028,"style":2968},[173,174],[39,3030,3032,3045],{"className":3031},[42],[39,3033,3035],{"className":3034},[46],[48,3036,3037],{"xmlns":50},[52,3038,3039,3043],{},[55,3040,3041],{},[155,3042,191],{},[62,3044,191],{"encoding":64},[39,3046,3048],{"className":3047,"ariaHidden":70},[69],[39,3049,3051,3054],{"className":3050},[74],[39,3052],{"className":3053,"style":169},[78],[39,3055,191],{"className":3056,"style":206},[173,174]," 之间找不到其他的理想","（除了 ",[39,3060,3062,3075],{"className":3061},[42],[39,3063,3065],{"className":3064},[46],[48,3066,3067],{"xmlns":50},[52,3068,3069,3073],{},[55,3070,3071],{},[155,3072,2953],{},[62,3074,2953],{"encoding":64},[39,3076,3078],{"className":3077,"ariaHidden":70},[69],[39,3079,3081,3084],{"className":3080},[74],[39,3082],{"className":3083,"style":169},[78],[39,3085,2953],{"className":3086,"style":2968},[173,174],[39,3088,3090,3103],{"className":3089},[42],[39,3091,3093],{"className":3092},[46],[48,3094,3095],{"xmlns":50},[52,3096,3097,3101],{},[55,3098,3099],{},[155,3100,191],{},[62,3102,191],{"encoding":64},[39,3104,3106],{"className":3105,"ariaHidden":70},[69],[39,3107,3109,3112],{"className":3108},[74],[39,3110],{"className":3111,"style":169},[78],[39,3113,191],{"className":3114,"style":206},[173,174]," 自己），则称 ",[39,3117,3119,3132],{"className":3118},[42],[39,3120,3122],{"className":3121},[46],[48,3123,3124],{"xmlns":50},[52,3125,3126,3130],{},[55,3127,3128],{},[155,3129,2953],{},[62,3131,2953],{"encoding":64},[39,3133,3135],{"className":3134,"ariaHidden":70},[69],[39,3136,3138,3141],{"className":3137},[74],[39,3139],{"className":3140,"style":169},[78],[39,3142,2953],{"className":3143,"style":2968},[173,174]," 为极大理想。",[28,3146,3147],{},[31,3148,3149,3150,1482,3226,3254,3255,753,3306,569],{},"数学语言：若 ",[39,3151,3153,3177],{"className":3152},[42],[39,3154,3156],{"className":3155},[46],[48,3157,3158],{"xmlns":50},[52,3159,3160,3174],{},[55,3161,3162,3164,3167,3170,3172],{},[155,3163,2953],{},[58,3165,3166],{},"⊆",[155,3168,3169],{},"I",[58,3171,3166],{},[155,3173,191],{},[62,3175,3176],{"encoding":64},"M \\subseteq I \\subseteq R",[39,3178,3180,3199,3217],{"className":3179,"ariaHidden":70},[69],[39,3181,3183,3187,3190,3193,3196],{"className":3182},[74],[39,3184],{"className":3185,"style":3186},[78],"height:0.8193em;vertical-align:-0.136em;",[39,3188,2953],{"className":3189,"style":2968},[173,174],[39,3191],{"className":3192,"style":248},[247],[39,3194,3166],{"className":3195},[83],[39,3197],{"className":3198,"style":248},[247],[39,3200,3202,3205,3208,3211,3214],{"className":3201},[74],[39,3203],{"className":3204,"style":3186},[78],[39,3206,3169],{"className":3207,"style":1098},[173,174],[39,3209],{"className":3210,"style":248},[247],[39,3212,3166],{"className":3213},[83],[39,3215],{"className":3216,"style":248},[247],[39,3218,3220,3223],{"className":3219},[74],[39,3221],{"className":3222,"style":169},[78],[39,3224,191],{"className":3225,"style":206},[173,174],[39,3227,3229,3242],{"className":3228},[42],[39,3230,3232],{"className":3231},[46],[48,3233,3234],{"xmlns":50},[52,3235,3236,3240],{},[55,3237,3238],{},[155,3239,3169],{},[62,3241,3169],{"encoding":64},[39,3243,3245],{"className":3244,"ariaHidden":70},[69],[39,3246,3248,3251],{"className":3247},[74],[39,3249],{"className":3250,"style":169},[78],[39,3252,3169],{"className":3253,"style":1098},[173,174]," 是理想，则必有 ",[39,3256,3258,3276],{"className":3257},[42],[39,3259,3261],{"className":3260},[46],[48,3262,3263],{"xmlns":50},[52,3264,3265,3273],{},[55,3266,3267,3269,3271],{},[155,3268,3169],{},[58,3270,292],{},[155,3272,2953],{},[62,3274,3275],{"encoding":64},"I=M",[39,3277,3279,3297],{"className":3278,"ariaHidden":70},[69],[39,3280,3282,3285,3288,3291,3294],{"className":3281},[74],[39,3283],{"className":3284,"style":169},[78],[39,3286,3169],{"className":3287,"style":1098},[173,174],[39,3289],{"className":3290,"style":248},[247],[39,3292,292],{"className":3293},[83],[39,3295],{"className":3296,"style":248},[247],[39,3298,3300,3303],{"className":3299},[74],[39,3301],{"className":3302,"style":169},[78],[39,3304,2953],{"className":3305,"style":2968},[173,174],[39,3307,3309,3327],{"className":3308},[42],[39,3310,3312],{"className":3311},[46],[48,3313,3314],{"xmlns":50},[52,3315,3316,3324],{},[55,3317,3318,3320,3322],{},[155,3319,3169],{},[58,3321,292],{},[155,3323,191],{},[62,3325,3326],{"encoding":64},"I=R",[39,3328,3330,3348],{"className":3329,"ariaHidden":70},[69],[39,3331,3333,3336,3339,3342,3345],{"className":3332},[74],[39,3334],{"className":3335,"style":169},[78],[39,3337,3169],{"className":3338,"style":1098},[173,174],[39,3340],{"className":3341,"style":248},[247],[39,3343,292],{"className":3344},[83],[39,3346],{"className":3347,"style":248},[247],[39,3349,3351,3354],{"className":3350},[74],[39,3352],{"className":3353,"style":169},[78],[39,3355,191],{"className":3356,"style":206},[173,174],[571,3358,3359],{},[20,3360,3361,821,3363,3391,3392,569],{},[24,3362,577],{},[39,3364,3366,3379],{"className":3365},[42],[39,3367,3369],{"className":3368},[46],[48,3370,3371],{"xmlns":50},[52,3372,3373,3377],{},[55,3374,3375],{},[155,3376,2953],{},[62,3378,2953],{"encoding":64},[39,3380,3382],{"className":3381,"ariaHidden":70},[69],[39,3383,3385,3388],{"className":3384},[74],[39,3386],{"className":3387,"style":169},[78],[39,3389,2953],{"className":3390,"style":2968},[173,174]," 是\"天花板\"级别的理想。它大到不能再大了，再往里面加任何一个不属于它的元素，它就会瞬间膨胀成整个环 ",[39,3393,3395,3408],{"className":3394},[42],[39,3396,3398],{"className":3397},[46],[48,3399,3400],{"xmlns":50},[52,3401,3402,3406],{},[55,3403,3404],{},[155,3405,191],{},[62,3407,191],{"encoding":64},[39,3409,3411],{"className":3410,"ariaHidden":70},[69],[39,3412,3414,3417],{"className":3413},[74],[39,3415],{"className":3416,"style":169},[78],[39,3418,191],{"className":3419,"style":206},[173,174],[135,3421,3423],{"id":3422},"_22-核心定理-b-重难点","2.2 核心定理 B (重难点)",[20,3425,3426,821,3429,3457,3458,894,3496,3535],{},[24,3427,3428],{},"定理 3.5.2",[39,3430,3432,3445],{"className":3431},[42],[39,3433,3435],{"className":3434},[46],[48,3436,3437],{"xmlns":50},[52,3438,3439,3443],{},[55,3440,3441],{},[155,3442,2953],{},[62,3444,2953],{"encoding":64},[39,3446,3448],{"className":3447,"ariaHidden":70},[69],[39,3449,3451,3454],{"className":3450},[74],[39,3452],{"className":3453,"style":169},[78],[39,3455,2953],{"className":3456,"style":2968},[173,174]," 是极大理想 ",[39,3459,3461,3478],{"className":3460},[42],[39,3462,3464],{"className":3463},[46],[48,3465,3466],{"xmlns":50},[52,3467,3468,3476],{},[55,3469,3470,3472,3474],{},[864,3471,866],{},[58,3473,869],{},[864,3475,866],{},[62,3477,874],{"encoding":64},[39,3479,3481],{"className":3480,"ariaHidden":70},[69],[39,3482,3484,3487,3490,3493],{"className":3483},[74],[39,3485],{"className":3486,"style":884},[78],[39,3488],{"className":3489,"style":248},[247],[39,3491,869],{"className":3492},[83],[39,3494],{"className":3495,"style":248},[247],[39,3497,3499,3517],{"className":3498},[42],[39,3500,3502],{"className":3501},[46],[48,3503,3504],{"xmlns":50},[52,3505,3506,3514],{},[55,3507,3508,3510,3512],{},[155,3509,191],{},[155,3511,911],{"mathvariant":224},[155,3513,2953],{},[62,3515,3516],{"encoding":64},"R\u002FM",[39,3518,3520],{"className":3519,"ariaHidden":70},[69],[39,3521,3523,3526,3529,3532],{"className":3522},[74],[39,3524],{"className":3525,"style":926},[78],[39,3527,191],{"className":3528,"style":206},[173,174],[39,3530,911],{"className":3531},[173],[39,3533,2953],{"className":3534,"style":2968},[173,174]," 是域。",[938,3537,940],{"id":3538},"证明详解-1",[20,3540,3541,3640],{},[24,3542,3543,3544,3572,3573,3601,3602,3535],{},"（",[39,3545,3547,3560],{"className":3546},[42],[39,3548,3550],{"className":3549},[46],[48,3551,3552],{"xmlns":50},[52,3553,3554,3558],{},[55,3555,3556],{},[58,3557,962],{},[62,3559,965],{"encoding":64},[39,3561,3563],{"className":3562,"ariaHidden":70},[69],[39,3564,3566,3569],{"className":3565},[74],[39,3567],{"className":3568,"style":975},[78],[39,3570,962],{"className":3571},[83],"）方向：已知 ",[39,3574,3576,3589],{"className":3575},[42],[39,3577,3579],{"className":3578},[46],[48,3580,3581],{"xmlns":50},[52,3582,3583,3587],{},[55,3584,3585],{},[155,3586,2953],{},[62,3588,2953],{"encoding":64},[39,3590,3592],{"className":3591,"ariaHidden":70},[69],[39,3593,3595,3598],{"className":3594},[74],[39,3596],{"className":3597,"style":169},[78],[39,3599,2953],{"className":3600,"style":2968},[173,174]," 极大，求证 ",[39,3603,3605,3622],{"className":3604},[42],[39,3606,3608],{"className":3607},[46],[48,3609,3610],{"xmlns":50},[52,3611,3612,3620],{},[55,3613,3614,3616,3618],{},[155,3615,191],{},[155,3617,911],{"mathvariant":224},[155,3619,2953],{},[62,3621,3516],{"encoding":64},[39,3623,3625],{"className":3624,"ariaHidden":70},[69],[39,3626,3628,3631,3634,3637],{"className":3627},[74],[39,3629],{"className":3630,"style":926},[78],[39,3632,191],{"className":3633,"style":206},[173,174],[39,3635,911],{"className":3636},[173],[39,3638,2953],{"className":3639,"style":2968},[173,174],[3641,3642,3643],"em",{},"关键思路：域就是“每个非零元素都有逆元”的环。我们要造出逆元。",[1048,3645,3646],{},[31,3647,3648,1244,3651,3689,3690,569],{},[24,3649,3650],{},"取元",[39,3652,3654,3671],{"className":3653},[42],[39,3655,3657],{"className":3656},[46],[48,3658,3659],{"xmlns":50},[52,3660,3661,3669],{},[55,3662,3663,3665,3667],{},[155,3664,191],{},[155,3666,911],{"mathvariant":224},[155,3668,2953],{},[62,3670,3516],{"encoding":64},[39,3672,3674],{"className":3673,"ariaHidden":70},[69],[39,3675,3677,3680,3683,3686],{"className":3676},[74],[39,3678],{"className":3679,"style":926},[78],[39,3681,191],{"className":3682,"style":206},[173,174],[39,3684,911],{"className":3685},[173],[39,3687,2953],{"className":3688,"style":2968},[173,174]," 中任取非零元素 ",[39,3691,3693,3711],{"className":3692},[42],[39,3694,3696],{"className":3695},[46],[48,3697,3698],{"xmlns":50},[52,3699,3700,3708],{},[55,3701,3702,3704,3706],{},[155,3703,320],{},[58,3705,1300],{},[155,3707,2953],{},[62,3709,3710],{"encoding":64},"a+M",[39,3712,3714,3732],{"className":3713,"ariaHidden":70},[69],[39,3715,3717,3720,3723,3726,3729],{"className":3716},[74],[39,3718],{"className":3719,"style":669},[78],[39,3721,320],{"className":3722},[173,174],[39,3724],{"className":3725,"style":676},[247],[39,3727,1300],{"className":3728},[680],[39,3730],{"className":3731,"style":676},[247],[39,3733,3735,3738],{"className":3734},[74],[39,3736],{"className":3737,"style":169},[78],[39,3739,2953],{"className":3740,"style":2968},[173,174],[28,3742,3743],{},[31,3744,1852,3745,569],{},[39,3746,3748,3766],{"className":3747},[42],[39,3749,3751],{"className":3750},[46],[48,3752,3753],{"xmlns":50},[52,3754,3755,3763],{},[55,3756,3757,3759,3761],{},[155,3758,320],{},[58,3760,1411],{"mathvariant":224},[155,3762,2953],{},[62,3764,3765],{"encoding":64},"a \\notin M",[39,3767,3769,3820],{"className":3768,"ariaHidden":70},[69],[39,3770,3772,3775,3778,3781,3817],{"className":3771},[74],[39,3773],{"className":3774,"style":926},[78],[39,3776,320],{"className":3777},[173,174],[39,3779],{"className":3780,"style":248},[247],[39,3782,3784,3790],{"className":3783},[83],[39,3785,3787],{"className":3786},[173],[39,3788,329],{"className":3789},[83],[39,3791,3793],{"className":3792},[173,258],[39,3794,3796],{"className":3795},[262],[39,3797,3799,3802,3814],{"className":3798},[1450],[39,3800],{"className":3801,"style":926},[78],[39,3803,3805],{"className":3804},[273],[39,3806,3808,3811],{"className":3807},[173],[39,3809,911],{"className":3810},[173],[39,3812],{"className":3813,"style":1466},[247],[39,3815],{"className":3816},[284],[39,3818],{"className":3819,"style":248},[247],[39,3821,3823,3826],{"className":3822},[74],[39,3824],{"className":3825,"style":169},[78],[39,3827,2953],{"className":3828,"style":2968},[173,174],[1048,3830,3832],{"start":3831},2,[31,3833,3834,3837,3838,1336,3866,3894,3895,569],{},[24,3835,3836],{},"构造新理想","：考虑由 ",[39,3839,3841,3854],{"className":3840},[42],[39,3842,3844],{"className":3843},[46],[48,3845,3846],{"xmlns":50},[52,3847,3848,3852],{},[55,3849,3850],{},[155,3851,2953],{},[62,3853,2953],{"encoding":64},[39,3855,3857],{"className":3856,"ariaHidden":70},[69],[39,3858,3860,3863],{"className":3859},[74],[39,3861],{"className":3862,"style":169},[78],[39,3864,2953],{"className":3865,"style":2968},[173,174],[39,3867,3869,3882],{"className":3868},[42],[39,3870,3872],{"className":3871},[46],[48,3873,3874],{"xmlns":50},[52,3875,3876,3880],{},[55,3877,3878],{},[155,3879,320],{},[62,3881,320],{"encoding":64},[39,3883,3885],{"className":3884,"ariaHidden":70},[69],[39,3886,3888,3891],{"className":3887},[74],[39,3889],{"className":3890,"style":749},[78],[39,3892,320],{"className":3893},[173,174]," 共同生成的理想，记为 ",[39,3896,3898,3922],{"className":3897},[42],[39,3899,3901],{"className":3900},[46],[48,3902,3903],{"xmlns":50},[52,3904,3905,3919],{},[55,3906,3907,3910,3912,3914,3916],{},[58,3908,3909],{"stretchy":1587},"⟨",[155,3911,2953],{},[58,3913,323],{"separator":70},[155,3915,320],{},[58,3917,3918],{"stretchy":1587},"⟩",[62,3920,3921],{"encoding":64},"\\langle M, a \\rangle",[39,3923,3925],{"className":3924,"ariaHidden":70},[69],[39,3926,3928,3931,3934,3937,3940,3943,3946],{"className":3927},[74],[39,3929],{"className":3930,"style":926},[78],[39,3932,3909],{"className":3933},[1637],[39,3935,2953],{"className":3936,"style":2968},[173,174],[39,3938,323],{"className":3939},[350],[39,3941],{"className":3942,"style":354},[247],[39,3944,320],{"className":3945},[173,174],[39,3947,3918],{"className":3948},[1662],[28,3950,3951],{},[31,3952,3953,3954,3982,3983,4011,4012,4040],{},"这个理想包含了 ",[39,3955,3957,3970],{"className":3956},[42],[39,3958,3960],{"className":3959},[46],[48,3961,3962],{"xmlns":50},[52,3963,3964,3968],{},[55,3965,3966],{},[155,3967,2953],{},[62,3969,2953],{"encoding":64},[39,3971,3973],{"className":3972,"ariaHidden":70},[69],[39,3974,3976,3979],{"className":3975},[74],[39,3977],{"className":3978,"style":169},[78],[39,3980,2953],{"className":3981,"style":2968},[173,174],"，而且因为有 ",[39,3984,3986,3999],{"className":3985},[42],[39,3987,3989],{"className":3988},[46],[48,3990,3991],{"xmlns":50},[52,3992,3993,3997],{},[55,3994,3995],{},[155,3996,320],{},[62,3998,320],{"encoding":64},[39,4000,4002],{"className":4001,"ariaHidden":70},[69],[39,4003,4005,4008],{"className":4004},[74],[39,4006],{"className":4007,"style":749},[78],[39,4009,320],{"className":4010},[173,174],"，它比 ",[39,4013,4015,4028],{"className":4014},[42],[39,4016,4018],{"className":4017},[46],[48,4019,4020],{"xmlns":50},[52,4021,4022,4026],{},[55,4023,4024],{},[155,4025,2953],{},[62,4027,2953],{"encoding":64},[39,4029,4031],{"className":4030,"ariaHidden":70},[69],[39,4032,4034,4037],{"className":4033},[74],[39,4035],{"className":4036,"style":169},[78],[39,4038,2953],{"className":4039,"style":2968},[173,174]," 更大。",[1048,4042,4044],{"start":4043},3,[31,4045,4046,1986,4049,4077,4078,4106,4107,4158,4159,4187,4188,569],{},[24,4047,4048],{},"利用极大性",[39,4050,4052,4065],{"className":4051},[42],[39,4053,4055],{"className":4054},[46],[48,4056,4057],{"xmlns":50},[52,4058,4059,4063],{},[55,4060,4061],{},[155,4062,2953],{},[62,4064,2953],{"encoding":64},[39,4066,4068],{"className":4067,"ariaHidden":70},[69],[39,4069,4071,4074],{"className":4070},[74],[39,4072],{"className":4073,"style":169},[78],[39,4075,2953],{"className":4076,"style":2968},[173,174]," 是极大理想，它上面只有 ",[39,4079,4081,4094],{"className":4080},[42],[39,4082,4084],{"className":4083},[46],[48,4085,4086],{"xmlns":50},[52,4087,4088,4092],{},[55,4089,4090],{},[155,4091,191],{},[62,4093,191],{"encoding":64},[39,4095,4097],{"className":4096,"ariaHidden":70},[69],[39,4098,4100,4103],{"className":4099},[74],[39,4101],{"className":4102,"style":169},[78],[39,4104,191],{"className":4105,"style":206},[173,174],"。既然 ",[39,4108,4110,4131],{"className":4109},[42],[39,4111,4113],{"className":4112},[46],[48,4114,4115],{"xmlns":50},[52,4116,4117,4129],{},[55,4118,4119,4121,4123,4125,4127],{},[58,4120,3909],{"stretchy":1587},[155,4122,2953],{},[58,4124,323],{"separator":70},[155,4126,320],{},[58,4128,3918],{"stretchy":1587},[62,4130,3921],{"encoding":64},[39,4132,4134],{"className":4133,"ariaHidden":70},[69],[39,4135,4137,4140,4143,4146,4149,4152,4155],{"className":4136},[74],[39,4138],{"className":4139,"style":926},[78],[39,4141,3909],{"className":4142},[1637],[39,4144,2953],{"className":4145,"style":2968},[173,174],[39,4147,323],{"className":4148},[350],[39,4150],{"className":4151,"style":354},[247],[39,4153,320],{"className":4154},[173,174],[39,4156,3918],{"className":4157},[1662]," 比 ",[39,4160,4162,4175],{"className":4161},[42],[39,4163,4165],{"className":4164},[46],[48,4166,4167],{"xmlns":50},[52,4168,4169,4173],{},[55,4170,4171],{},[155,4172,2953],{},[62,4174,2953],{"encoding":64},[39,4176,4178],{"className":4177,"ariaHidden":70},[69],[39,4179,4181,4184],{"className":4180},[74],[39,4182],{"className":4183,"style":169},[78],[39,4185,2953],{"className":4186,"style":2968},[173,174]," 大，那它",[24,4189,4190,4191],{},"只能是 ",[39,4192,4194,4207],{"className":4193},[42],[39,4195,4197],{"className":4196},[46],[48,4198,4199],{"xmlns":50},[52,4200,4201,4205],{},[55,4202,4203],{},[155,4204,191],{},[62,4206,191],{"encoding":64},[39,4208,4210],{"className":4209,"ariaHidden":70},[69],[39,4211,4213,4216],{"className":4212},[74],[39,4214],{"className":4215,"style":169},[78],[39,4217,191],{"className":4218,"style":206},[173,174],[28,4220,4221],{},[31,4222,4223,821,4226,569],{},[24,4224,4225],{},"结论",[39,4227,4229,4255],{"className":4228},[42],[39,4230,4232],{"className":4231},[46],[48,4233,4234],{"xmlns":50},[52,4235,4236,4252],{},[55,4237,4238,4240,4242,4244,4246,4248,4250],{},[58,4239,3909],{"stretchy":1587},[155,4241,2953],{},[58,4243,323],{"separator":70},[155,4245,320],{},[58,4247,3918],{"stretchy":1587},[58,4249,292],{},[155,4251,191],{},[62,4253,4254],{"encoding":64},"\\langle M, a \\rangle = R",[39,4256,4258,4291],{"className":4257,"ariaHidden":70},[69],[39,4259,4261,4264,4267,4270,4273,4276,4279,4282,4285,4288],{"className":4260},[74],[39,4262],{"className":4263,"style":926},[78],[39,4265,3909],{"className":4266},[1637],[39,4268,2953],{"className":4269,"style":2968},[173,174],[39,4271,323],{"className":4272},[350],[39,4274],{"className":4275,"style":354},[247],[39,4277,320],{"className":4278},[173,174],[39,4280,3918],{"className":4281},[1662],[39,4283],{"className":4284,"style":248},[247],[39,4286,292],{"className":4287},[83],[39,4289],{"className":4290,"style":248},[247],[39,4292,4294,4297],{"className":4293},[74],[39,4295],{"className":4296,"style":169},[78],[39,4298,191],{"className":4299,"style":206},[173,174],[1048,4301,4303,4414],{"start":4302},4,[31,4304,4305,4308,4309,4337,4338,569],{},[24,4306,4307],{},"抓取单位元","：既然整个环 ",[39,4310,4312,4325],{"className":4311},[42],[39,4313,4315],{"className":4314},[46],[48,4316,4317],{"xmlns":50},[52,4318,4319,4323],{},[55,4320,4321],{},[155,4322,191],{},[62,4324,191],{"encoding":64},[39,4326,4328],{"className":4327,"ariaHidden":70},[69],[39,4329,4331,4334],{"className":4330},[74],[39,4332],{"className":4333,"style":169},[78],[39,4335,191],{"className":4336,"style":206},[173,174]," 都在这个理想里，那么单位元 ",[39,4339,4341,4368],{"className":4340},[42],[39,4342,4344],{"className":4343},[46],[48,4345,4346],{"xmlns":50},[52,4347,4348,4365],{},[55,4349,4350,4353,4355,4357,4359,4361,4363],{},[1080,4351,4352],{},"1",[58,4354,329],{},[58,4356,3909],{"stretchy":1587},[155,4358,2953],{},[58,4360,323],{"separator":70},[155,4362,320],{},[58,4364,3918],{"stretchy":1587},[62,4366,4367],{"encoding":64},"1 \\in \\langle M, a \\rangle",[39,4369,4371,4390],{"className":4370,"ariaHidden":70},[69],[39,4372,4374,4378,4381,4384,4387],{"className":4373},[74],[39,4375],{"className":4376,"style":4377},[78],"height:0.6835em;vertical-align:-0.0391em;",[39,4379,4352],{"className":4380},[173],[39,4382],{"className":4383,"style":248},[247],[39,4385,329],{"className":4386},[83],[39,4388],{"className":4389,"style":248},[247],[39,4391,4393,4396,4399,4402,4405,4408,4411],{"className":4392},[74],[39,4394],{"className":4395,"style":926},[78],[39,4397,3909],{"className":4398},[1637],[39,4400,2953],{"className":4401,"style":2968},[173,174],[39,4403,323],{"className":4404},[350],[39,4406],{"className":4407,"style":354},[247],[39,4409,320],{"className":4410},[173,174],[39,4412,3918],{"className":4413},[1662],[31,4415,4416,821,4419,4447,4448,4476,4477,4505],{},[24,4417,4418],{},"写出表达式",[39,4420,4422,4435],{"className":4421},[42],[39,4423,4425],{"className":4424},[46],[48,4426,4427],{"xmlns":50},[52,4428,4429,4433],{},[55,4430,4431],{},[1080,4432,4352],{},[62,4434,4352],{"encoding":64},[39,4436,4438],{"className":4437,"ariaHidden":70},[69],[39,4439,4441,4444],{"className":4440},[74],[39,4442],{"className":4443,"style":1132},[78],[39,4445,4352],{"className":4446},[173]," 可以写成 ",[39,4449,4451,4464],{"className":4450},[42],[39,4452,4454],{"className":4453},[46],[48,4455,4456],{"xmlns":50},[52,4457,4458,4462],{},[55,4459,4460],{},[155,4461,2953],{},[62,4463,2953],{"encoding":64},[39,4465,4467],{"className":4466,"ariaHidden":70},[69],[39,4468,4470,4473],{"className":4469},[74],[39,4471],{"className":4472,"style":169},[78],[39,4474,2953],{"className":4475,"style":2968},[173,174]," 里的元素和 ",[39,4478,4480,4493],{"className":4479},[42],[39,4481,4483],{"className":4482},[46],[48,4484,4485],{"xmlns":50},[52,4486,4487,4491],{},[55,4488,4489],{},[155,4490,320],{},[62,4492,320],{"encoding":64},[39,4494,4496],{"className":4495,"ariaHidden":70},[69],[39,4497,4499,4502],{"className":4498},[74],[39,4500],{"className":4501,"style":749},[78],[39,4503,320],{"className":4504},[173,174]," 的倍数之和。",[28,4507,4508],{},[31,4509,4510,4511,1336,4563,4616,4617,569],{},"存在 ",[39,4512,4514,4533],{"className":4513},[42],[39,4515,4517],{"className":4516},[46],[48,4518,4519],{"xmlns":50},[52,4520,4521,4530],{},[55,4522,4523,4526,4528],{},[155,4524,4525],{},"m",[58,4527,329],{},[155,4529,2953],{},[62,4531,4532],{"encoding":64},"m \\in M",[39,4534,4536,4554],{"className":4535,"ariaHidden":70},[69],[39,4537,4539,4542,4545,4548,4551],{"className":4538},[74],[39,4540],{"className":4541,"style":463},[78],[39,4543,4525],{"className":4544},[173,174],[39,4546],{"className":4547,"style":248},[247],[39,4549,329],{"className":4550},[83],[39,4552],{"className":4553,"style":248},[247],[39,4555,4557,4560],{"className":4556},[74],[39,4558],{"className":4559,"style":169},[78],[39,4561,2953],{"className":4562,"style":2968},[173,174],[39,4564,4566,4585],{"className":4565},[42],[39,4567,4569],{"className":4568},[46],[48,4570,4571],{"xmlns":50},[52,4572,4573,4582],{},[55,4574,4575,4578,4580],{},[155,4576,4577],{},"r",[58,4579,329],{},[155,4581,191],{},[62,4583,4584],{"encoding":64},"r \\in R",[39,4586,4588,4607],{"className":4587,"ariaHidden":70},[69],[39,4589,4591,4594,4598,4601,4604],{"className":4590},[74],[39,4592],{"className":4593,"style":463},[78],[39,4595,4577],{"className":4596,"style":4597},[173,174],"margin-right:0.0278em;",[39,4599],{"className":4600,"style":248},[247],[39,4602,329],{"className":4603},[83],[39,4605],{"className":4606,"style":248},[247],[39,4608,4610,4613],{"className":4609},[74],[39,4611],{"className":4612,"style":169},[78],[39,4614,191],{"className":4615,"style":206},[173,174],"，使得 ",[39,4618,4620,4644],{"className":4619},[42],[39,4621,4623],{"className":4622},[46],[48,4624,4625],{"xmlns":50},[52,4626,4627,4641],{},[55,4628,4629,4631,4633,4635,4637,4639],{},[1080,4630,4352],{},[58,4632,292],{},[155,4634,4525],{},[58,4636,1300],{},[155,4638,4577],{},[155,4640,320],{},[62,4642,4643],{"encoding":64},"1 = m + ra",[39,4645,4647,4665,4683],{"className":4646,"ariaHidden":70},[69],[39,4648,4650,4653,4656,4659,4662],{"className":4649},[74],[39,4651],{"className":4652,"style":1132},[78],[39,4654,4352],{"className":4655},[173],[39,4657],{"className":4658,"style":248},[247],[39,4660,292],{"className":4661},[83],[39,4663],{"className":4664,"style":248},[247],[39,4666,4668,4671,4674,4677,4680],{"className":4667},[74],[39,4669],{"className":4670,"style":669},[78],[39,4672,4525],{"className":4673},[173,174],[39,4675],{"className":4676,"style":676},[247],[39,4678,1300],{"className":4679},[680],[39,4681],{"className":4682,"style":676},[247],[39,4684,4686,4689,4692],{"className":4685},[74],[39,4687],{"className":4688,"style":749},[78],[39,4690,4577],{"className":4691,"style":4597},[173,174],[39,4693,320],{"className":4694},[173,174],[1048,4696,4698],{"start":4697},6,[31,4699,4700,4703,4704,4732],{},[24,4701,4702],{},"回到商环","：两边同时模 ",[39,4705,4707,4720],{"className":4706},[42],[39,4708,4710],{"className":4709},[46],[48,4711,4712],{"xmlns":50},[52,4713,4714,4718],{},[55,4715,4716],{},[155,4717,2953],{},[62,4719,2953],{"encoding":64},[39,4721,4723],{"className":4722,"ariaHidden":70},[69],[39,4724,4726,4729],{"className":4725},[74],[39,4727],{"className":4728,"style":169},[78],[39,4730,2953],{"className":4731,"style":2968},[173,174],"（即在商环中看等式）：",[28,4734,4735,4869,4923],{},[31,4736,4737],{},[39,4738,4740,4776],{"className":4739},[42],[39,4741,4743],{"className":4742},[46],[48,4744,4745],{"xmlns":50},[52,4746,4747,4773],{},[55,4748,4749,4751,4753,4755,4757,4759,4761,4763,4765,4767,4769,4771],{},[1080,4750,4352],{},[58,4752,1300],{},[155,4754,2953],{},[58,4756,292],{},[58,4758,947],{"stretchy":1587},[155,4760,4525],{},[58,4762,1300],{},[155,4764,4577],{},[155,4766,320],{},[58,4768,1596],{"stretchy":1587},[58,4770,1300],{},[155,4772,2953],{},[62,4774,4775],{"encoding":64},"1 + M = (m + ra) + M",[39,4777,4779,4797,4815,4836,4860],{"className":4778,"ariaHidden":70},[69],[39,4780,4782,4785,4788,4791,4794],{"className":4781},[74],[39,4783],{"className":4784,"style":1810},[78],[39,4786,4352],{"className":4787},[173],[39,4789],{"className":4790,"style":676},[247],[39,4792,1300],{"className":4793},[680],[39,4795],{"className":4796,"style":676},[247],[39,4798,4800,4803,4806,4809,4812],{"className":4799},[74],[39,4801],{"className":4802,"style":169},[78],[39,4804,2953],{"className":4805,"style":2968},[173,174],[39,4807],{"className":4808,"style":248},[247],[39,4810,292],{"className":4811},[83],[39,4813],{"className":4814,"style":248},[247],[39,4816,4818,4821,4824,4827,4830,4833],{"className":4817},[74],[39,4819],{"className":4820,"style":926},[78],[39,4822,947],{"className":4823},[1637],[39,4825,4525],{"className":4826},[173,174],[39,4828],{"className":4829,"style":676},[247],[39,4831,1300],{"className":4832},[680],[39,4834],{"className":4835,"style":676},[247],[39,4837,4839,4842,4845,4848,4851,4854,4857],{"className":4838},[74],[39,4840],{"className":4841,"style":926},[78],[39,4843,4577],{"className":4844,"style":4597},[173,174],[39,4846,320],{"className":4847},[173,174],[39,4849,1596],{"className":4850},[1662],[39,4852],{"className":4853,"style":676},[247],[39,4855,1300],{"className":4856},[680],[39,4858],{"className":4859,"style":676},[247],[39,4861,4863,4866],{"className":4862},[74],[39,4864],{"className":4865,"style":169},[78],[39,4867,2953],{"className":4868,"style":2968},[173,174],[31,4870,4871,4872,4922],{},"因为 ",[39,4873,4875,4892],{"className":4874},[42],[39,4876,4878],{"className":4877},[46],[48,4879,4880],{"xmlns":50},[52,4881,4882,4890],{},[55,4883,4884,4886,4888],{},[155,4885,4525],{},[58,4887,329],{},[155,4889,2953],{},[62,4891,4532],{"encoding":64},[39,4893,4895,4913],{"className":4894,"ariaHidden":70},[69],[39,4896,4898,4901,4904,4907,4910],{"className":4897},[74],[39,4899],{"className":4900,"style":463},[78],[39,4902,4525],{"className":4903},[173,174],[39,4905],{"className":4906,"style":248},[247],[39,4908,329],{"className":4909},[83],[39,4911],{"className":4912,"style":248},[247],[39,4914,4916,4919],{"className":4915},[74],[39,4917],{"className":4918,"style":169},[78],[39,4920,2953],{"className":4921,"style":2968},[173,174],"，它在商环里是 0。",[31,4924,4925,4926,569],{},"所以 ",[39,4927,4929,4979],{"className":4928},[42],[39,4930,4932],{"className":4931},[46],[48,4933,4934],{"xmlns":50},[52,4935,4936,4976],{},[55,4937,4938,4940,4942,4944,4946,4948,4950,4952,4954,4956,4958,4960,4962,4964,4966,4968,4970,4972,4974],{},[1080,4939,4352],{},[58,4941,1300],{},[155,4943,2953],{},[58,4945,292],{},[155,4947,4577],{},[155,4949,320],{},[58,4951,1300],{},[155,4953,2953],{},[58,4955,292],{},[58,4957,947],{"stretchy":1587},[155,4959,4577],{},[58,4961,1300],{},[155,4963,2953],{},[58,4965,1596],{"stretchy":1587},[58,4967,947],{"stretchy":1587},[155,4969,320],{},[58,4971,1300],{},[155,4973,2953],{},[58,4975,1596],{"stretchy":1587},[62,4977,4978],{"encoding":64},"1 + M = ra + M = (r+M)(a+M)",[39,4980,4982,5000,5018,5039,5057,5078,5105],{"className":4981,"ariaHidden":70},[69],[39,4983,4985,4988,4991,4994,4997],{"className":4984},[74],[39,4986],{"className":4987,"style":1810},[78],[39,4989,4352],{"className":4990},[173],[39,4992],{"className":4993,"style":676},[247],[39,4995,1300],{"className":4996},[680],[39,4998],{"className":4999,"style":676},[247],[39,5001,5003,5006,5009,5012,5015],{"className":5002},[74],[39,5004],{"className":5005,"style":169},[78],[39,5007,2953],{"className":5008,"style":2968},[173,174],[39,5010],{"className":5011,"style":248},[247],[39,5013,292],{"className":5014},[83],[39,5016],{"className":5017,"style":248},[247],[39,5019,5021,5024,5027,5030,5033,5036],{"className":5020},[74],[39,5022],{"className":5023,"style":669},[78],[39,5025,4577],{"className":5026,"style":4597},[173,174],[39,5028,320],{"className":5029},[173,174],[39,5031],{"className":5032,"style":676},[247],[39,5034,1300],{"className":5035},[680],[39,5037],{"className":5038,"style":676},[247],[39,5040,5042,5045,5048,5051,5054],{"className":5041},[74],[39,5043],{"className":5044,"style":169},[78],[39,5046,2953],{"className":5047,"style":2968},[173,174],[39,5049],{"className":5050,"style":248},[247],[39,5052,292],{"className":5053},[83],[39,5055],{"className":5056,"style":248},[247],[39,5058,5060,5063,5066,5069,5072,5075],{"className":5059},[74],[39,5061],{"className":5062,"style":926},[78],[39,5064,947],{"className":5065},[1637],[39,5067,4577],{"className":5068,"style":4597},[173,174],[39,5070],{"className":5071,"style":676},[247],[39,5073,1300],{"className":5074},[680],[39,5076],{"className":5077,"style":676},[247],[39,5079,5081,5084,5087,5090,5093,5096,5099,5102],{"className":5080},[74],[39,5082],{"className":5083,"style":926},[78],[39,5085,2953],{"className":5086,"style":2968},[173,174],[39,5088,1596],{"className":5089},[1662],[39,5091,947],{"className":5092},[1637],[39,5094,320],{"className":5095},[173,174],[39,5097],{"className":5098,"style":676},[247],[39,5100,1300],{"className":5101},[680],[39,5103],{"className":5104,"style":676},[247],[39,5106,5108,5111,5114],{"className":5107},[74],[39,5109],{"className":5110,"style":926},[78],[39,5112,2953],{"className":5113,"style":2968},[173,174],[39,5115,1596],{"className":5116},[1662],[1048,5118,5120],{"start":5119},7,[31,5121,5122,821,5124,5174,5175,5226,5227,5255,5256,5306,5307,3535],{},[24,5123,2201],{},[39,5125,5127,5144],{"className":5126},[42],[39,5128,5130],{"className":5129},[46],[48,5131,5132],{"xmlns":50},[52,5133,5134,5142],{},[55,5135,5136,5138,5140],{},[155,5137,320],{},[58,5139,1300],{},[155,5141,2953],{},[62,5143,3710],{"encoding":64},[39,5145,5147,5165],{"className":5146,"ariaHidden":70},[69],[39,5148,5150,5153,5156,5159,5162],{"className":5149},[74],[39,5151],{"className":5152,"style":669},[78],[39,5154,320],{"className":5155},[173,174],[39,5157],{"className":5158,"style":676},[247],[39,5160,1300],{"className":5161},[680],[39,5163],{"className":5164,"style":676},[247],[39,5166,5168,5171],{"className":5167},[74],[39,5169],{"className":5170,"style":169},[78],[39,5172,2953],{"className":5173,"style":2968},[173,174]," 乘以 ",[39,5176,5178,5196],{"className":5177},[42],[39,5179,5181],{"className":5180},[46],[48,5182,5183],{"xmlns":50},[52,5184,5185,5193],{},[55,5186,5187,5189,5191],{},[155,5188,4577],{},[58,5190,1300],{},[155,5192,2953],{},[62,5194,5195],{"encoding":64},"r+M",[39,5197,5199,5217],{"className":5198,"ariaHidden":70},[69],[39,5200,5202,5205,5208,5211,5214],{"className":5201},[74],[39,5203],{"className":5204,"style":669},[78],[39,5206,4577],{"className":5207,"style":4597},[173,174],[39,5209],{"className":5210,"style":676},[247],[39,5212,1300],{"className":5213},[680],[39,5215],{"className":5216,"style":676},[247],[39,5218,5220,5223],{"className":5219},[74],[39,5221],{"className":5222,"style":169},[78],[39,5224,2953],{"className":5225,"style":2968},[173,174]," 等于 ",[39,5228,5230,5243],{"className":5229},[42],[39,5231,5233],{"className":5232},[46],[48,5234,5235],{"xmlns":50},[52,5236,5237,5241],{},[55,5238,5239],{},[1080,5240,4352],{},[62,5242,4352],{"encoding":64},[39,5244,5246],{"className":5245,"ariaHidden":70},[69],[39,5247,5249,5252],{"className":5248},[74],[39,5250],{"className":5251,"style":1132},[78],[39,5253,4352],{"className":5254},[173],"，说明 ",[39,5257,5259,5276],{"className":5258},[42],[39,5260,5262],{"className":5261},[46],[48,5263,5264],{"xmlns":50},[52,5265,5266,5274],{},[55,5267,5268,5270,5272],{},[155,5269,320],{},[58,5271,1300],{},[155,5273,2953],{},[62,5275,3710],{"encoding":64},[39,5277,5279,5297],{"className":5278,"ariaHidden":70},[69],[39,5280,5282,5285,5288,5291,5294],{"className":5281},[74],[39,5283],{"className":5284,"style":669},[78],[39,5286,320],{"className":5287},[173,174],[39,5289],{"className":5290,"style":676},[247],[39,5292,1300],{"className":5293},[680],[39,5295],{"className":5296,"style":676},[247],[39,5298,5300,5303],{"className":5299},[74],[39,5301],{"className":5302,"style":169},[78],[39,5304,2953],{"className":5305,"style":2968},[173,174]," 有逆元。所以 ",[39,5308,5310,5327],{"className":5309},[42],[39,5311,5313],{"className":5312},[46],[48,5314,5315],{"xmlns":50},[52,5316,5317,5325],{},[55,5318,5319,5321,5323],{},[155,5320,191],{},[155,5322,911],{"mathvariant":224},[155,5324,2953],{},[62,5326,3516],{"encoding":64},[39,5328,5330],{"className":5329,"ariaHidden":70},[69],[39,5331,5333,5336,5339,5342],{"className":5332},[74],[39,5334],{"className":5335,"style":926},[78],[39,5337,191],{"className":5338,"style":206},[173,174],[39,5340,911],{"className":5341},[173],[39,5343,2953],{"className":5344,"style":2968},[173,174],[20,5346,5347],{},[24,5348,3543,5349,3572,5377,5415,5416,5444],{},[39,5350,5352,5365],{"className":5351},[42],[39,5353,5355],{"className":5354},[46],[48,5356,5357],{"xmlns":50},[52,5358,5359,5363],{},[55,5360,5361],{},[58,5362,2260],{},[62,5364,2263],{"encoding":64},[39,5366,5368],{"className":5367,"ariaHidden":70},[69],[39,5369,5371,5374],{"className":5370},[74],[39,5372],{"className":5373,"style":975},[78],[39,5375,2260],{"className":5376},[83],[39,5378,5380,5397],{"className":5379},[42],[39,5381,5383],{"className":5382},[46],[48,5384,5385],{"xmlns":50},[52,5386,5387,5395],{},[55,5388,5389,5391,5393],{},[155,5390,191],{},[155,5392,911],{"mathvariant":224},[155,5394,2953],{},[62,5396,3516],{"encoding":64},[39,5398,5400],{"className":5399,"ariaHidden":70},[69],[39,5401,5403,5406,5409,5412],{"className":5402},[74],[39,5404],{"className":5405,"style":926},[78],[39,5407,191],{"className":5408,"style":206},[173,174],[39,5410,911],{"className":5411},[173],[39,5413,2953],{"className":5414,"style":2968},[173,174]," 是域，求证 ",[39,5417,5419,5432],{"className":5418},[42],[39,5420,5422],{"className":5421},[46],[48,5423,5424],{"xmlns":50},[52,5425,5426,5430],{},[55,5427,5428],{},[155,5429,2953],{},[62,5431,2953],{"encoding":64},[39,5433,5435],{"className":5434,"ariaHidden":70},[69],[39,5436,5438,5441],{"className":5437},[74],[39,5439],{"className":5440,"style":169},[78],[39,5442,2953],{"className":5443,"style":2968},[173,174]," 极大。",[1048,5446,5447,5639,5890],{},[31,5448,5449,5452,5453,5481,5482,5554,5555,569],{},[24,5450,5451],{},"假设中间层","：假设有一个理想 ",[39,5454,5456,5469],{"className":5455},[42],[39,5457,5459],{"className":5458},[46],[48,5460,5461],{"xmlns":50},[52,5462,5463,5467],{},[55,5464,5465],{},[155,5466,3169],{},[62,5468,3169],{"encoding":64},[39,5470,5472],{"className":5471,"ariaHidden":70},[69],[39,5473,5475,5478],{"className":5474},[74],[39,5476],{"className":5477,"style":169},[78],[39,5479,3169],{"className":5480,"style":1098},[173,174]," 夹在中间：",[39,5483,5485,5506],{"className":5484},[42],[39,5486,5488],{"className":5487},[46],[48,5489,5490],{"xmlns":50},[52,5491,5492,5504],{},[55,5493,5494,5496,5498,5500,5502],{},[155,5495,2953],{},[58,5497,3166],{},[155,5499,3169],{},[58,5501,3166],{},[155,5503,191],{},[62,5505,3176],{"encoding":64},[39,5507,5509,5527,5545],{"className":5508,"ariaHidden":70},[69],[39,5510,5512,5515,5518,5521,5524],{"className":5511},[74],[39,5513],{"className":5514,"style":3186},[78],[39,5516,2953],{"className":5517,"style":2968},[173,174],[39,5519],{"className":5520,"style":248},[247],[39,5522,3166],{"className":5523},[83],[39,5525],{"className":5526,"style":248},[247],[39,5528,5530,5533,5536,5539,5542],{"className":5529},[74],[39,5531],{"className":5532,"style":3186},[78],[39,5534,3169],{"className":5535,"style":1098},[173,174],[39,5537],{"className":5538,"style":248},[247],[39,5540,3166],{"className":5541},[83],[39,5543],{"className":5544,"style":248},[247],[39,5546,5548,5551],{"className":5547},[74],[39,5549],{"className":5550,"style":169},[78],[39,5552,191],{"className":5553,"style":206},[173,174],"，且 ",[39,5556,5558,5576],{"className":5557},[42],[39,5559,5561],{"className":5560},[46],[48,5562,5563],{"xmlns":50},[52,5564,5565,5573],{},[55,5566,5567,5569,5571],{},[155,5568,2953],{},[58,5570,225],{"mathvariant":224},[155,5572,3169],{},[62,5574,5575],{"encoding":64},"M \\neq I",[39,5577,5579,5630],{"className":5578,"ariaHidden":70},[69],[39,5580,5582,5585,5588,5591,5627],{"className":5581},[74],[39,5583],{"className":5584,"style":240},[78],[39,5586,2953],{"className":5587,"style":2968},[173,174],[39,5589],{"className":5590,"style":248},[247],[39,5592,5594,5621,5624],{"className":5593},[83],[39,5595,5597],{"className":5596},[83],[39,5598,5600],{"className":5599},[173,258],[39,5601,5603],{"className":5602},[262],[39,5604,5606,5609,5618],{"className":5605},[266],[39,5607],{"className":5608,"style":240},[78],[39,5610,5612],{"className":5611},[273],[39,5613,5615],{"className":5614},[173],[39,5616,280],{"className":5617},[83],[39,5619],{"className":5620},[284],[39,5622],{"className":5623},[247,288],[39,5625,292],{"className":5626},[83],[39,5628],{"className":5629,"style":248},[247],[39,5631,5633,5636],{"className":5632},[74],[39,5634],{"className":5635,"style":169},[78],[39,5637,3169],{"className":5638,"style":1098},[173,174],[31,5640,5641,1986,5644,5727,5728,5756,5757,5785,5786,5814,5815,5889],{},[24,5642,5643],{},"取差值",[39,5645,5647,5664],{"className":5646},[42],[39,5648,5650],{"className":5649},[46],[48,5651,5652],{"xmlns":50},[52,5653,5654,5662],{},[55,5655,5656,5658,5660],{},[155,5657,2953],{},[58,5659,225],{"mathvariant":224},[155,5661,3169],{},[62,5663,5575],{"encoding":64},[39,5665,5667,5718],{"className":5666,"ariaHidden":70},[69],[39,5668,5670,5673,5676,5679,5715],{"className":5669},[74],[39,5671],{"className":5672,"style":240},[78],[39,5674,2953],{"className":5675,"style":2968},[173,174],[39,5677],{"className":5678,"style":248},[247],[39,5680,5682,5709,5712],{"className":5681},[83],[39,5683,5685],{"className":5684},[83],[39,5686,5688],{"className":5687},[173,258],[39,5689,5691],{"className":5690},[262],[39,5692,5694,5697,5706],{"className":5693},[266],[39,5695],{"className":5696,"style":240},[78],[39,5698,5700],{"className":5699},[273],[39,5701,5703],{"className":5702},[173],[39,5704,280],{"className":5705},[83],[39,5707],{"className":5708},[284],[39,5710],{"className":5711},[247,288],[39,5713,292],{"className":5714},[83],[39,5716],{"className":5717,"style":248},[247],[39,5719,5721,5724],{"className":5720},[74],[39,5722],{"className":5723,"style":169},[78],[39,5725,3169],{"className":5726,"style":1098},[173,174],"，必定存在一个元素 ",[39,5729,5731,5744],{"className":5730},[42],[39,5732,5734],{"className":5733},[46],[48,5735,5736],{"xmlns":50},[52,5737,5738,5742],{},[55,5739,5740],{},[155,5741,320],{},[62,5743,320],{"encoding":64},[39,5745,5747],{"className":5746,"ariaHidden":70},[69],[39,5748,5750,5753],{"className":5749},[74],[39,5751],{"className":5752,"style":749},[78],[39,5754,320],{"className":5755},[173,174],"，它在 ",[39,5758,5760,5773],{"className":5759},[42],[39,5761,5763],{"className":5762},[46],[48,5764,5765],{"xmlns":50},[52,5766,5767,5771],{},[55,5768,5769],{},[155,5770,3169],{},[62,5772,3169],{"encoding":64},[39,5774,5776],{"className":5775,"ariaHidden":70},[69],[39,5777,5779,5782],{"className":5778},[74],[39,5780],{"className":5781,"style":169},[78],[39,5783,3169],{"className":5784,"style":1098},[173,174]," 里但不在 ",[39,5787,5789,5802],{"className":5788},[42],[39,5790,5792],{"className":5791},[46],[48,5793,5794],{"xmlns":50},[52,5795,5796,5800],{},[55,5797,5798],{},[155,5799,2953],{},[62,5801,2953],{"encoding":64},[39,5803,5805],{"className":5804,"ariaHidden":70},[69],[39,5806,5808,5811],{"className":5807},[74],[39,5809],{"className":5810,"style":169},[78],[39,5812,2953],{"className":5813,"style":2968},[173,174]," 里（",[39,5816,5818,5841],{"className":5817},[42],[39,5819,5821],{"className":5820},[46],[48,5822,5823],{"xmlns":50},[52,5824,5825,5838],{},[55,5826,5827,5829,5831,5833,5836],{},[155,5828,320],{},[58,5830,329],{},[155,5832,3169],{},[58,5834,5835],{},"∖",[155,5837,2953],{},[62,5839,5840],{"encoding":64},"a \\in I \\setminus M",[39,5842,5844,5862,5880],{"className":5843,"ariaHidden":70},[69],[39,5845,5847,5850,5853,5856,5859],{"className":5846},[74],[39,5848],{"className":5849,"style":463},[78],[39,5851,320],{"className":5852},[173,174],[39,5854],{"className":5855,"style":248},[247],[39,5857,329],{"className":5858},[83],[39,5860],{"className":5861,"style":248},[247],[39,5863,5865,5868,5871,5874,5877],{"className":5864},[74],[39,5866],{"className":5867,"style":926},[78],[39,5869,3169],{"className":5870,"style":1098},[173,174],[39,5872],{"className":5873,"style":676},[247],[39,5875,5835],{"className":5876},[680],[39,5878],{"className":5879,"style":676},[247],[39,5881,5883,5886],{"className":5882},[74],[39,5884],{"className":5885,"style":169},[78],[39,5887,2953],{"className":5888,"style":2968},[173,174],"）。",[31,5891,5892,1244,5895,5933,5934,5984,5985,6023,6024,6074,6075,569],{},[24,5893,5894],{},"利用域的性质",[39,5896,5898,5915],{"className":5897},[42],[39,5899,5901],{"className":5900},[46],[48,5902,5903],{"xmlns":50},[52,5904,5905,5913],{},[55,5906,5907,5909,5911],{},[155,5908,191],{},[155,5910,911],{"mathvariant":224},[155,5912,2953],{},[62,5914,3516],{"encoding":64},[39,5916,5918],{"className":5917,"ariaHidden":70},[69],[39,5919,5921,5924,5927,5930],{"className":5920},[74],[39,5922],{"className":5923,"style":926},[78],[39,5925,191],{"className":5926,"style":206},[173,174],[39,5928,911],{"className":5929},[173],[39,5931,2953],{"className":5932,"style":2968},[173,174]," 中，",[39,5935,5937,5954],{"className":5936},[42],[39,5938,5940],{"className":5939},[46],[48,5941,5942],{"xmlns":50},[52,5943,5944,5952],{},[55,5945,5946,5948,5950],{},[155,5947,320],{},[58,5949,1300],{},[155,5951,2953],{},[62,5953,3710],{"encoding":64},[39,5955,5957,5975],{"className":5956,"ariaHidden":70},[69],[39,5958,5960,5963,5966,5969,5972],{"className":5959},[74],[39,5961],{"className":5962,"style":669},[78],[39,5964,320],{"className":5965},[173,174],[39,5967],{"className":5968,"style":676},[247],[39,5970,1300],{"className":5971},[680],[39,5973],{"className":5974,"style":676},[247],[39,5976,5978,5981],{"className":5977},[74],[39,5979],{"className":5980,"style":169},[78],[39,5982,2953],{"className":5983,"style":2968},[173,174]," 是非零的。因为 ",[39,5986,5988,6005],{"className":5987},[42],[39,5989,5991],{"className":5990},[46],[48,5992,5993],{"xmlns":50},[52,5994,5995,6003],{},[55,5996,5997,5999,6001],{},[155,5998,191],{},[155,6000,911],{"mathvariant":224},[155,6002,2953],{},[62,6004,3516],{"encoding":64},[39,6006,6008],{"className":6007,"ariaHidden":70},[69],[39,6009,6011,6014,6017,6020],{"className":6010},[74],[39,6012],{"className":6013,"style":926},[78],[39,6015,191],{"className":6016,"style":206},[173,174],[39,6018,911],{"className":6019},[173],[39,6021,2953],{"className":6022,"style":2968},[173,174]," 是域，所以 ",[39,6025,6027,6044],{"className":6026},[42],[39,6028,6030],{"className":6029},[46],[48,6031,6032],{"xmlns":50},[52,6033,6034,6042],{},[55,6035,6036,6038,6040],{},[155,6037,320],{},[58,6039,1300],{},[155,6041,2953],{},[62,6043,3710],{"encoding":64},[39,6045,6047,6065],{"className":6046,"ariaHidden":70},[69],[39,6048,6050,6053,6056,6059,6062],{"className":6049},[74],[39,6051],{"className":6052,"style":669},[78],[39,6054,320],{"className":6055},[173,174],[39,6057],{"className":6058,"style":676},[247],[39,6060,1300],{"className":6061},[680],[39,6063],{"className":6064,"style":676},[247],[39,6066,6068,6071],{"className":6067},[74],[39,6069],{"className":6070,"style":169},[78],[39,6072,2953],{"className":6073,"style":2968},[173,174]," 有逆元，设为 ",[39,6076,6078,6096],{"className":6077},[42],[39,6079,6081],{"className":6080},[46],[48,6082,6083],{"xmlns":50},[52,6084,6085,6093],{},[55,6086,6087,6089,6091],{},[155,6088,326],{},[58,6090,1300],{},[155,6092,2953],{},[62,6094,6095],{"encoding":64},"b+M",[39,6097,6099,6117],{"className":6098,"ariaHidden":70},[69],[39,6100,6102,6105,6108,6111,6114],{"className":6101},[74],[39,6103],{"className":6104,"style":1367},[78],[39,6106,326],{"className":6107},[173,174],[39,6109],{"className":6110,"style":676},[247],[39,6112,1300],{"className":6113},[680],[39,6115],{"className":6116,"style":676},[247],[39,6118,6120,6123],{"className":6119},[74],[39,6121],{"className":6122,"style":169},[78],[39,6124,2953],{"className":6125,"style":2968},[173,174],[28,6127,6128],{},[31,6129,6130,6131,569],{},"即 ",[39,6132,6134,6194],{"className":6133},[42],[39,6135,6137],{"className":6136},[46],[48,6138,6139],{"xmlns":50},[52,6140,6141,6191],{},[55,6142,6143,6145,6147,6149,6151,6153,6155,6157,6159,6161,6163,6165,6167,6169,6171,6173,6176,6178,6180,6182,6185,6187,6189],{},[58,6144,947],{"stretchy":1587},[155,6146,320],{},[58,6148,1300],{},[155,6150,2953],{},[58,6152,1596],{"stretchy":1587},[58,6154,947],{"stretchy":1587},[155,6156,326],{},[58,6158,1300],{},[155,6160,2953],{},[58,6162,1596],{"stretchy":1587},[58,6164,292],{},[1080,6166,4352],{},[58,6168,1300],{},[155,6170,2953],{},[864,6172,866],{},[58,6174,6175],{},"⟹",[864,6177,866],{},[155,6179,320],{},[155,6181,326],{},[58,6183,6184],{},"−",[1080,6186,4352],{},[58,6188,329],{},[155,6190,2953],{},[62,6192,6193],{"encoding":64},"(a+M)(b+M) = 1+M \\implies ab - 1 \\in M",[39,6195,6197,6218,6245,6266,6284,6309,6327,6345],{"className":6196,"ariaHidden":70},[69],[39,6198,6200,6203,6206,6209,6212,6215],{"className":6199},[74],[39,6201],{"className":6202,"style":926},[78],[39,6204,947],{"className":6205},[1637],[39,6207,320],{"className":6208},[173,174],[39,6210],{"className":6211,"style":676},[247],[39,6213,1300],{"className":6214},[680],[39,6216],{"className":6217,"style":676},[247],[39,6219,6221,6224,6227,6230,6233,6236,6239,6242],{"className":6220},[74],[39,6222],{"className":6223,"style":926},[78],[39,6225,2953],{"className":6226,"style":2968},[173,174],[39,6228,1596],{"className":6229},[1662],[39,6231,947],{"className":6232},[1637],[39,6234,326],{"className":6235},[173,174],[39,6237],{"className":6238,"style":676},[247],[39,6240,1300],{"className":6241},[680],[39,6243],{"className":6244,"style":676},[247],[39,6246,6248,6251,6254,6257,6260,6263],{"className":6247},[74],[39,6249],{"className":6250,"style":926},[78],[39,6252,2953],{"className":6253,"style":2968},[173,174],[39,6255,1596],{"className":6256},[1662],[39,6258],{"className":6259,"style":248},[247],[39,6261,292],{"className":6262},[83],[39,6264],{"className":6265,"style":248},[247],[39,6267,6269,6272,6275,6278,6281],{"className":6268},[74],[39,6270],{"className":6271,"style":1810},[78],[39,6273,4352],{"className":6274},[173],[39,6276],{"className":6277,"style":676},[247],[39,6279,1300],{"className":6280},[680],[39,6282],{"className":6283,"style":676},[247],[39,6285,6287,6291,6294,6297,6300,6303,6306],{"className":6286},[74],[39,6288],{"className":6289,"style":6290},[78],"height:0.7073em;vertical-align:-0.024em;",[39,6292,2953],{"className":6293,"style":2968},[173,174],[39,6295],{"className":6296,"style":248},[247],[39,6298],{"className":6299,"style":248},[247],[39,6301,6175],{"className":6302},[83],[39,6304],{"className":6305,"style":248},[247],[39,6307],{"className":6308,"style":248},[247],[39,6310,6312,6315,6318,6321,6324],{"className":6311},[74],[39,6313],{"className":6314,"style":1367},[78],[39,6316,413],{"className":6317},[173,174],[39,6319],{"className":6320,"style":676},[247],[39,6322,6184],{"className":6323},[680],[39,6325],{"className":6326,"style":676},[247],[39,6328,6330,6333,6336,6339,6342],{"className":6329},[74],[39,6331],{"className":6332,"style":4377},[78],[39,6334,4352],{"className":6335},[173],[39,6337],{"className":6338,"style":248},[247],[39,6340,329],{"className":6341},[83],[39,6343],{"className":6344,"style":248},[247],[39,6346,6348,6351],{"className":6347},[74],[39,6349],{"className":6350,"style":169},[78],[39,6352,2953],{"className":6353,"style":2968},[173,174],[1048,6355,6356],{"start":4302},[31,6357,6358,821],{},[24,6359,6360],{},"推导",[28,6362,6363,6492],{},[31,6364,6365,6366,6441,6442,5889],{},"记 ",[39,6367,6369,6393],{"className":6368},[42],[39,6370,6372],{"className":6371},[46],[48,6373,6374],{"xmlns":50},[52,6375,6376,6390],{},[55,6377,6378,6380,6382,6384,6386,6388],{},[155,6379,320],{},[155,6381,326],{},[58,6383,6184],{},[1080,6385,4352],{},[58,6387,292],{},[155,6389,4525],{},[62,6391,6392],{"encoding":64},"ab - 1 = m",[39,6394,6396,6414,6432],{"className":6395,"ariaHidden":70},[69],[39,6397,6399,6402,6405,6408,6411],{"className":6398},[74],[39,6400],{"className":6401,"style":1367},[78],[39,6403,413],{"className":6404},[173,174],[39,6406],{"className":6407,"style":676},[247],[39,6409,6184],{"className":6410},[680],[39,6412],{"className":6413,"style":676},[247],[39,6415,6417,6420,6423,6426,6429],{"className":6416},[74],[39,6418],{"className":6419,"style":1132},[78],[39,6421,4352],{"className":6422},[173],[39,6424],{"className":6425,"style":248},[247],[39,6427,292],{"className":6428},[83],[39,6430],{"className":6431,"style":248},[247],[39,6433,6435,6438],{"className":6434},[74],[39,6436],{"className":6437,"style":749},[78],[39,6439,4525],{"className":6440},[173,174],"（其中 ",[39,6443,6445,6462],{"className":6444},[42],[39,6446,6448],{"className":6447},[46],[48,6449,6450],{"xmlns":50},[52,6451,6452,6460],{},[55,6453,6454,6456,6458],{},[155,6455,4525],{},[58,6457,329],{},[155,6459,2953],{},[62,6461,4532],{"encoding":64},[39,6463,6465,6483],{"className":6464,"ariaHidden":70},[69],[39,6466,6468,6471,6474,6477,6480],{"className":6467},[74],[39,6469],{"className":6470,"style":463},[78],[39,6472,4525],{"className":6473},[173,174],[39,6475],{"className":6476,"style":248},[247],[39,6478,329],{"className":6479},[83],[39,6481],{"className":6482,"style":248},[247],[39,6484,6486,6489],{"className":6485},[74],[39,6487],{"className":6488,"style":169},[78],[39,6490,2953],{"className":6491,"style":2968},[173,174],[31,6493,6494,6495,569],{},"移项得：",[39,6496,6498,6522],{"className":6497},[42],[39,6499,6501],{"className":6500},[46],[48,6502,6503],{"xmlns":50},[52,6504,6505,6519],{},[55,6506,6507,6509,6511,6513,6515,6517],{},[1080,6508,4352],{},[58,6510,292],{},[155,6512,320],{},[155,6514,326],{},[58,6516,6184],{},[155,6518,4525],{},[62,6520,6521],{"encoding":64},"1 = ab - m",[39,6523,6525,6543,6561],{"className":6524,"ariaHidden":70},[69],[39,6526,6528,6531,6534,6537,6540],{"className":6527},[74],[39,6529],{"className":6530,"style":1132},[78],[39,6532,4352],{"className":6533},[173],[39,6535],{"className":6536,"style":248},[247],[39,6538,292],{"className":6539},[83],[39,6541],{"className":6542,"style":248},[247],[39,6544,6546,6549,6552,6555,6558],{"className":6545},[74],[39,6547],{"className":6548,"style":1367},[78],[39,6550,413],{"className":6551},[173,174],[39,6553],{"className":6554,"style":676},[247],[39,6556,6184],{"className":6557},[680],[39,6559],{"className":6560,"style":676},[247],[39,6562,6564,6567],{"className":6563},[74],[39,6565],{"className":6566,"style":749},[78],[39,6568,4525],{"className":6569},[173,174],[1048,6571,6573],{"start":6572},5,[31,6574,6575,821],{},[24,6576,6577],{},"判断归属",[28,6579,6580,6690,6821,6936],{},[31,6581,6582,6689],{},[39,6583,6585,6617],{"className":6584},[42],[39,6586,6588],{"className":6587},[46],[48,6589,6590],{"xmlns":50},[52,6591,6592,6614],{},[55,6593,6594,6596,6598,6600,6602,6604,6606,6608,6610,6612],{},[155,6595,320],{},[58,6597,329],{},[155,6599,3169],{},[864,6601,866],{},[58,6603,6175],{},[864,6605,866],{},[155,6607,320],{},[155,6609,326],{},[58,6611,329],{},[155,6613,3169],{},[62,6615,6616],{"encoding":64},"a \\in I \\implies ab \\in I",[39,6618,6620,6638,6662,6680],{"className":6619,"ariaHidden":70},[69],[39,6621,6623,6626,6629,6632,6635],{"className":6622},[74],[39,6624],{"className":6625,"style":463},[78],[39,6627,320],{"className":6628},[173,174],[39,6630],{"className":6631,"style":248},[247],[39,6633,329],{"className":6634},[83],[39,6636],{"className":6637,"style":248},[247],[39,6639,6641,6644,6647,6650,6653,6656,6659],{"className":6640},[74],[39,6642],{"className":6643,"style":6290},[78],[39,6645,3169],{"className":6646,"style":1098},[173,174],[39,6648],{"className":6649,"style":248},[247],[39,6651],{"className":6652,"style":248},[247],[39,6654,6175],{"className":6655},[83],[39,6657],{"className":6658,"style":248},[247],[39,6660],{"className":6661,"style":248},[247],[39,6663,6665,6668,6671,6674,6677],{"className":6664},[74],[39,6666],{"className":6667,"style":409},[78],[39,6669,413],{"className":6670},[173,174],[39,6672],{"className":6673,"style":248},[247],[39,6675,329],{"className":6676},[83],[39,6678],{"className":6679,"style":248},[247],[39,6681,6683,6686],{"className":6682},[74],[39,6684],{"className":6685,"style":169},[78],[39,6687,3169],{"className":6688,"style":1098},[173,174],"（理想吸收律）。",[31,6691,6692,569],{},[39,6693,6695,6730],{"className":6694},[42],[39,6696,6698],{"className":6697},[46],[48,6699,6700],{"xmlns":50},[52,6701,6702,6727],{},[55,6703,6704,6706,6708,6710,6713,6715,6717,6719,6721,6723,6725],{},[155,6705,4525],{},[58,6707,329],{},[155,6709,2953],{},[58,6711,6712],{},"⊂",[155,6714,3169],{},[864,6716,866],{},[58,6718,6175],{},[864,6720,866],{},[155,6722,4525],{},[58,6724,329],{},[155,6726,3169],{},[62,6728,6729],{"encoding":64},"m \\in M \\subset I \\implies m \\in I",[39,6731,6733,6751,6770,6794,6812],{"className":6732,"ariaHidden":70},[69],[39,6734,6736,6739,6742,6745,6748],{"className":6735},[74],[39,6737],{"className":6738,"style":463},[78],[39,6740,4525],{"className":6741},[173,174],[39,6743],{"className":6744,"style":248},[247],[39,6746,329],{"className":6747},[83],[39,6749],{"className":6750,"style":248},[247],[39,6752,6754,6758,6761,6764,6767],{"className":6753},[74],[39,6755],{"className":6756,"style":6757},[78],"height:0.7224em;vertical-align:-0.0391em;",[39,6759,2953],{"className":6760,"style":2968},[173,174],[39,6762],{"className":6763,"style":248},[247],[39,6765,6712],{"className":6766},[83],[39,6768],{"className":6769,"style":248},[247],[39,6771,6773,6776,6779,6782,6785,6788,6791],{"className":6772},[74],[39,6774],{"className":6775,"style":6290},[78],[39,6777,3169],{"className":6778,"style":1098},[173,174],[39,6780],{"className":6781,"style":248},[247],[39,6783],{"className":6784,"style":248},[247],[39,6786,6175],{"className":6787},[83],[39,6789],{"className":6790,"style":248},[247],[39,6792],{"className":6793,"style":248},[247],[39,6795,6797,6800,6803,6806,6809],{"className":6796},[74],[39,6798],{"className":6799,"style":463},[78],[39,6801,4525],{"className":6802},[173,174],[39,6804],{"className":6805,"style":248},[247],[39,6807,329],{"className":6808},[83],[39,6810],{"className":6811,"style":248},[247],[39,6813,6815,6818],{"className":6814},[74],[39,6816],{"className":6817,"style":169},[78],[39,6819,3169],{"className":6820,"style":1098},[173,174],[31,6822,6823,6824,6852,6853,6906,6907,6935],{},"两个都在 ",[39,6825,6827,6840],{"className":6826},[42],[39,6828,6830],{"className":6829},[46],[48,6831,6832],{"xmlns":50},[52,6833,6834,6838],{},[55,6835,6836],{},[155,6837,3169],{},[62,6839,3169],{"encoding":64},[39,6841,6843],{"className":6842,"ariaHidden":70},[69],[39,6844,6846,6849],{"className":6845},[74],[39,6847],{"className":6848,"style":169},[78],[39,6850,3169],{"className":6851,"style":1098},[173,174]," 里，它们的差 ",[39,6854,6856,6876],{"className":6855},[42],[39,6857,6859],{"className":6858},[46],[48,6860,6861],{"xmlns":50},[52,6862,6863,6873],{},[55,6864,6865,6867,6869,6871],{},[155,6866,320],{},[155,6868,326],{},[58,6870,6184],{},[155,6872,4525],{},[62,6874,6875],{"encoding":64},"ab - m",[39,6877,6879,6897],{"className":6878,"ariaHidden":70},[69],[39,6880,6882,6885,6888,6891,6894],{"className":6881},[74],[39,6883],{"className":6884,"style":1367},[78],[39,6886,413],{"className":6887},[173,174],[39,6889],{"className":6890,"style":676},[247],[39,6892,6184],{"className":6893},[680],[39,6895],{"className":6896,"style":676},[247],[39,6898,6900,6903],{"className":6899},[74],[39,6901],{"className":6902,"style":749},[78],[39,6904,4525],{"className":6905},[173,174]," 也在 ",[39,6908,6910,6923],{"className":6909},[42],[39,6911,6913],{"className":6912},[46],[48,6914,6915],{"xmlns":50},[52,6916,6917,6921],{},[55,6918,6919],{},[155,6920,3169],{},[62,6922,3169],{"encoding":64},[39,6924,6926],{"className":6925,"ariaHidden":70},[69],[39,6927,6929,6932],{"className":6928},[74],[39,6930],{"className":6931,"style":169},[78],[39,6933,3169],{"className":6934,"style":1098},[173,174]," 里。",[31,6937,4925,6938,569],{},[39,6939,6941,6959],{"className":6940},[42],[39,6942,6944],{"className":6943},[46],[48,6945,6946],{"xmlns":50},[52,6947,6948,6956],{},[55,6949,6950,6952,6954],{},[1080,6951,4352],{},[58,6953,329],{},[155,6955,3169],{},[62,6957,6958],{"encoding":64},"1 \\in I",[39,6960,6962,6980],{"className":6961,"ariaHidden":70},[69],[39,6963,6965,6968,6971,6974,6977],{"className":6964},[74],[39,6966],{"className":6967,"style":4377},[78],[39,6969,4352],{"className":6970},[173],[39,6972],{"className":6973,"style":248},[247],[39,6975,329],{"className":6976},[83],[39,6978],{"className":6979,"style":248},[247],[39,6981,6983,6986],{"className":6982},[74],[39,6984],{"className":6985,"style":169},[78],[39,6987,3169],{"className":6988,"style":1098},[173,174],[1048,6990,6991,7105],{"start":4697},[31,6992,6993,6995,6996,7024,7025,7053,7054,569],{},[24,6994,4225],{},"：如果理想 ",[39,6997,6999,7012],{"className":6998},[42],[39,7000,7002],{"className":7001},[46],[48,7003,7004],{"xmlns":50},[52,7005,7006,7010],{},[55,7007,7008],{},[155,7009,3169],{},[62,7011,3169],{"encoding":64},[39,7013,7015],{"className":7014,"ariaHidden":70},[69],[39,7016,7018,7021],{"className":7017},[74],[39,7019],{"className":7020,"style":169},[78],[39,7022,3169],{"className":7023,"style":1098},[173,174]," 含有 ",[39,7026,7028,7041],{"className":7027},[42],[39,7029,7031],{"className":7030},[46],[48,7032,7033],{"xmlns":50},[52,7034,7035,7039],{},[55,7036,7037],{},[1080,7038,4352],{},[62,7040,4352],{"encoding":64},[39,7042,7044],{"className":7043,"ariaHidden":70},[69],[39,7045,7047,7050],{"className":7046},[74],[39,7048],{"className":7049,"style":1132},[78],[39,7051,4352],{"className":7052},[173],"，那么 ",[39,7055,7057,7075],{"className":7056},[42],[39,7058,7060],{"className":7059},[46],[48,7061,7062],{"xmlns":50},[52,7063,7064,7072],{},[55,7065,7066,7068,7070],{},[155,7067,3169],{},[58,7069,292],{},[155,7071,191],{},[62,7073,7074],{"encoding":64},"I = R",[39,7076,7078,7096],{"className":7077,"ariaHidden":70},[69],[39,7079,7081,7084,7087,7090,7093],{"className":7080},[74],[39,7082],{"className":7083,"style":169},[78],[39,7085,3169],{"className":7086,"style":1098},[173,174],[39,7088],{"className":7089,"style":248},[247],[39,7091,292],{"className":7092},[83],[39,7094],{"className":7095,"style":248},[247],[39,7097,7099,7102],{"className":7098},[74],[39,7100],{"className":7101,"style":169},[78],[39,7103,191],{"className":7104,"style":206},[173,174],[31,7106,7107,7109,7110,7138,7139,7167,7168,7196],{},[24,7108,2201],{},"：夹在中间的理想 ",[39,7111,7113,7126],{"className":7112},[42],[39,7114,7116],{"className":7115},[46],[48,7117,7118],{"xmlns":50},[52,7119,7120,7124],{},[55,7121,7122],{},[155,7123,3169],{},[62,7125,3169],{"encoding":64},[39,7127,7129],{"className":7128,"ariaHidden":70},[69],[39,7130,7132,7135],{"className":7131},[74],[39,7133],{"className":7134,"style":169},[78],[39,7136,3169],{"className":7137,"style":1098},[173,174]," 只能是 ",[39,7140,7142,7155],{"className":7141},[42],[39,7143,7145],{"className":7144},[46],[48,7146,7147],{"xmlns":50},[52,7148,7149,7153],{},[55,7150,7151],{},[155,7152,191],{},[62,7154,191],{"encoding":64},[39,7156,7158],{"className":7157,"ariaHidden":70},[69],[39,7159,7161,7164],{"className":7160},[74],[39,7162],{"className":7163,"style":169},[78],[39,7165,191],{"className":7166,"style":206},[173,174],"，所以 ",[39,7169,7171,7184],{"className":7170},[42],[39,7172,7174],{"className":7173},[46],[48,7175,7176],{"xmlns":50},[52,7177,7178,7182],{},[55,7179,7180],{},[155,7181,2953],{},[62,7183,2953],{"encoding":64},[39,7185,7187],{"className":7186,"ariaHidden":70},[69],[39,7188,7190,7193],{"className":7189},[74],[39,7191],{"className":7192,"style":169},[78],[39,7194,2953],{"className":7195,"style":2968},[173,174]," 是极大的。",[128,7198],{},[15,7200,7202],{"id":7201},"_3-两者关系与重要推论","3. 两者关系与重要推论",[28,7204,7205,7298],{},[31,7206,7207,7210,7211,7254,7255,7297],{},[24,7208,7209],{},"逻辑链","：\n因为",[24,7212,7213,7214,7253],{},"域 ",[39,7215,7217,7235],{"className":7216},[42],[39,7218,7220],{"className":7219},[46],[48,7221,7222],{"xmlns":50},[52,7223,7224,7232],{},[55,7225,7226,7228,7230],{},[864,7227,866],{},[58,7229,6175],{},[864,7231,866],{},[62,7233,7234],{"encoding":64},"\\implies",[39,7236,7238],{"className":7237,"ariaHidden":70},[69],[39,7239,7241,7244,7247,7250],{"className":7240},[74],[39,7242],{"className":7243,"style":884},[78],[39,7245],{"className":7246,"style":248},[247],[39,7248,6175],{"className":7249},[83],[39,7251],{"className":7252,"style":248},[247]," 整环","，所以",[24,7256,7257,7258,7296],{},"极大理想 ",[39,7259,7261,7278],{"className":7260},[42],[39,7262,7264],{"className":7263},[46],[48,7265,7266],{"xmlns":50},[52,7267,7268,7276],{},[55,7269,7270,7272,7274],{},[864,7271,866],{},[58,7273,6175],{},[864,7275,866],{},[62,7277,7234],{"encoding":64},[39,7279,7281],{"className":7280,"ariaHidden":70},[69],[39,7282,7284,7287,7290,7293],{"className":7283},[74],[39,7285],{"className":7286,"style":884},[78],[39,7288],{"className":7289,"style":248},[247],[39,7291,6175],{"className":7292},[83],[39,7294],{"className":7295,"style":248},[247]," 素理想","。\n（口诀：极大的肯定素，素的不一定大。）",[31,7299,7300,7303,7304],{},[24,7301,7302],{},"存在性（Zorn 引理）","：\n",[28,7305,7306,7312],{},[31,7307,7308,7311],{},[24,7309,7310],{},"推论 3.5.4","：任何带单位元的交换环，一定至少有一个极大理想。",[31,7313,7314],{},"证明用到了佐恩引理，核心思想是\"不断把理想变大，直到撞到天花板\"。",[128,7316],{},[15,7318,7320],{"id":7319},"_4-例子","4. 例子",[7322,7323,7324,7437],"table",{},[7325,7326,7327],"thead",{},[7328,7329,7330,7362,7393,7435],"tr",{},[7331,7332,7333,7334],"th",{},"环 ",[39,7335,7337,7350],{"className":7336},[42],[39,7338,7340],{"className":7339},[46],[48,7341,7342],{"xmlns":50},[52,7343,7344,7348],{},[55,7345,7346],{},[155,7347,191],{},[62,7349,191],{"encoding":64},[39,7351,7353],{"className":7352,"ariaHidden":70},[69],[39,7354,7356,7359],{"className":7355},[74],[39,7357],{"className":7358,"style":169},[78],[39,7360,191],{"className":7361,"style":206},[173,174],[7331,7363,7364,7365],{},"理想 ",[39,7366,7368,7381],{"className":7367},[42],[39,7369,7371],{"className":7370},[46],[48,7372,7373],{"xmlns":50},[52,7374,7375,7379],{},[55,7376,7377],{},[155,7378,3169],{},[62,7380,3169],{"encoding":64},[39,7382,7384],{"className":7383,"ariaHidden":70},[69],[39,7385,7387,7390],{"className":7386},[74],[39,7388],{"className":7389,"style":169},[78],[39,7391,3169],{"className":7392,"style":1098},[173,174],[7331,7394,7395,7434],{},[39,7396,7398,7416],{"className":7397},[42],[39,7399,7401],{"className":7400},[46],[48,7402,7403],{"xmlns":50},[52,7404,7405,7413],{},[55,7406,7407,7409,7411],{},[155,7408,191],{},[155,7410,911],{"mathvariant":224},[155,7412,3169],{},[62,7414,7415],{"encoding":64},"R\u002FI",[39,7417,7419],{"className":7418,"ariaHidden":70},[69],[39,7420,7422,7425,7428,7431],{"className":7421},[74],[39,7423],{"className":7424,"style":926},[78],[39,7426,191],{"className":7427,"style":206},[173,174],[39,7429,911],{"className":7430},[173],[39,7432,3169],{"className":7433,"style":1098},[173,174]," 是什么？",[7331,7436,4225],{},[7438,7439,7440,7660,7809,8103],"tbody",{},[7328,7441,7442,7481,7547,7624],{},[7443,7444,7445],"td",{},[24,7446,7447,7448],{},"整数环 ",[39,7449,7451,7467],{"className":7450},[42],[39,7452,7454],{"className":7453},[46],[48,7455,7456],{"xmlns":50},[52,7457,7458,7464],{},[55,7459,7460],{},[155,7461,7463],{"mathvariant":7462},"double-struck","Z",[62,7465,7466],{"encoding":64},"\\mathbb{Z}",[39,7468,7470],{"className":7469,"ariaHidden":70},[69],[39,7471,7473,7477],{"className":7472},[74],[39,7474],{"className":7475,"style":7476},[78],"height:0.6889em;",[39,7478,7463],{"className":7479},[173,7480],"mathbb",[7443,7482,7483,3543,7518,7546],{},[39,7484,7486,7502],{"className":7485},[42],[39,7487,7489],{"className":7488},[46],[48,7490,7491],{"xmlns":50},[52,7492,7493,7499],{},[55,7494,7495,7497],{},[155,7496,20],{},[155,7498,7463],{"mathvariant":7462},[62,7500,7501],{"encoding":64},"p\\mathbb{Z}",[39,7503,7505],{"className":7504,"ariaHidden":70},[69],[39,7506,7508,7512,7515],{"className":7507},[74],[39,7509],{"className":7510,"style":7511},[78],"height:0.8833em;vertical-align:-0.1944em;",[39,7513,20],{"className":7514},[173,174],[39,7516,7463],{"className":7517},[173,7480],[39,7519,7521,7534],{"className":7520},[42],[39,7522,7524],{"className":7523},[46],[48,7525,7526],{"xmlns":50},[52,7527,7528,7532],{},[55,7529,7530],{},[155,7531,20],{},[62,7533,20],{"encoding":64},[39,7535,7537],{"className":7536,"ariaHidden":70},[69],[39,7538,7540,7543],{"className":7539},[74],[39,7541],{"className":7542,"style":604},[78],[39,7544,20],{"className":7545},[173,174]," 是素数）",[7443,7548,7549,7623],{},[39,7550,7552,7570],{"className":7551},[42],[39,7553,7555],{"className":7554},[46],[48,7556,7557],{"xmlns":50},[52,7558,7559,7567],{},[55,7560,7561],{},[1610,7562,7563,7565],{},[155,7564,7463],{"mathvariant":7462},[155,7566,20],{},[62,7568,7569],{"encoding":64},"\\mathbb{Z}_p",[39,7571,7573],{"className":7572,"ariaHidden":70},[69],[39,7574,7576,7580],{"className":7575},[74],[39,7577],{"className":7578,"style":7579},[78],"height:0.975em;vertical-align:-0.2861em;",[39,7581,7583,7586],{"className":7582},[173],[39,7584,7463],{"className":7585},[173,7480],[39,7587,7589],{"className":7588},[1715],[39,7590,7592,7614],{"className":7591},[1719,1720],[39,7593,7595,7611],{"className":7594},[1724],[39,7596,7599],{"className":7597,"style":7598},[1728],"height:0.1514em;",[39,7600,7602,7605],{"style":7601},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[39,7603],{"className":7604,"style":1737},[1736],[39,7606,7608],{"className":7607},[1741,1742,1743,1744],[39,7609,20],{"className":7610},[173,174,1744],[39,7612,1761],{"className":7613},[1760],[39,7615,7617],{"className":7616},[1724],[39,7618,7621],{"className":7619,"style":7620},[1728],"height:0.2861em;",[39,7622],{},"（有限域）",[7443,7625,7626,7659],{},[39,7627,7629,7644],{"className":7628},[42],[39,7630,7632],{"className":7631},[46],[48,7633,7634],{"xmlns":50},[52,7635,7636,7642],{},[55,7637,7638,7640],{},[155,7639,20],{},[155,7641,7463],{"mathvariant":7462},[62,7643,7501],{"encoding":64},[39,7645,7647],{"className":7646,"ariaHidden":70},[69],[39,7648,7650,7653,7656],{"className":7649},[74],[39,7651],{"className":7652,"style":7511},[78],[39,7654,20],{"className":7655},[173,174],[39,7657,7463],{"className":7658},[173,7480]," 是极大理想（也是素理想）",[7328,7661,7662,7694,7737,7768],{},[7443,7663,7664],{},[24,7665,7447,7666],{},[39,7667,7669,7682],{"className":7668},[42],[39,7670,7672],{"className":7671},[46],[48,7673,7674],{"xmlns":50},[52,7675,7676,7680],{},[55,7677,7678],{},[155,7679,7463],{"mathvariant":7462},[62,7681,7466],{"encoding":64},[39,7683,7685],{"className":7684,"ariaHidden":70},[69],[39,7686,7688,7691],{"className":7687},[74],[39,7689],{"className":7690,"style":7476},[78],[39,7692,7463],{"className":7693},[173,7480],[7443,7695,7696],{},[39,7697,7699,7719],{"className":7698},[42],[39,7700,7702],{"className":7701},[46],[48,7703,7704],{"xmlns":50},[52,7705,7706,7716],{},[55,7707,7708,7711,7713],{},[58,7709,7710],{"stretchy":1587},"{",[1080,7712,1082],{},[58,7714,7715],{"stretchy":1587},"}",[62,7717,7718],{"encoding":64},"\\{0\\}",[39,7720,7722],{"className":7721,"ariaHidden":70},[69],[39,7723,7725,7728,7731,7734],{"className":7724},[74],[39,7726],{"className":7727,"style":926},[78],[39,7729,7710],{"className":7730},[1637],[39,7732,1082],{"className":7733},[173],[39,7735,7715],{"className":7736},[1662],[7443,7738,7739,7767],{},[39,7740,7742,7755],{"className":7741},[42],[39,7743,7745],{"className":7744},[46],[48,7746,7747],{"xmlns":50},[52,7748,7749,7753],{},[55,7750,7751],{},[155,7752,7463],{"mathvariant":7462},[62,7754,7466],{"encoding":64},[39,7756,7758],{"className":7757,"ariaHidden":70},[69],[39,7759,7761,7764],{"className":7760},[74],[39,7762],{"className":7763,"style":7476},[78],[39,7765,7463],{"className":7766},[173,7480],"（是整环但不是域）",[7443,7769,7770,7808],{},[39,7771,7773,7790],{"className":7772},[42],[39,7774,7776],{"className":7775},[46],[48,7777,7778],{"xmlns":50},[52,7779,7780,7788],{},[55,7781,7782,7784,7786],{},[58,7783,7710],{"stretchy":1587},[1080,7785,1082],{},[58,7787,7715],{"stretchy":1587},[62,7789,7718],{"encoding":64},[39,7791,7793],{"className":7792,"ariaHidden":70},[69],[39,7794,7796,7799,7802,7805],{"className":7795},[74],[39,7797],{"className":7798,"style":926},[78],[39,7800,7710],{"className":7801},[1637],[39,7803,1082],{"className":7804},[173],[39,7806,7715],{"className":7807},[1662]," 是素理想，但不是极大理想",[7328,7810,7811,7863,7961,8010],{},[7443,7812,7813],{},[24,7814,7815,7816],{},"实多项式 ",[39,7817,7819,7842],{"className":7818},[42],[39,7820,7822],{"className":7821},[46],[48,7823,7824],{"xmlns":50},[52,7825,7826,7839],{},[55,7827,7828,7830,7833,7836],{},[155,7829,191],{"mathvariant":7462},[58,7831,7832],{"stretchy":1587},"[",[155,7834,7835],{},"x",[58,7837,7838],{"stretchy":1587},"]",[62,7840,7841],{"encoding":64},"\\mathbb{R}[x]",[39,7843,7845],{"className":7844,"ariaHidden":70},[69],[39,7846,7848,7851,7854,7857,7860],{"className":7847},[74],[39,7849],{"className":7850,"style":926},[78],[39,7852,191],{"className":7853},[173,7480],[39,7855,7832],{"className":7856},[1637],[39,7858,7835],{"className":7859},[173,174],[39,7861,7838],{"className":7862},[1662],[7443,7864,7865],{},[39,7866,7868,7896],{"className":7867},[42],[39,7869,7871],{"className":7870},[46],[48,7872,7873],{"xmlns":50},[52,7874,7875,7893],{},[55,7876,7877,7879,7887,7889,7891],{},[58,7878,3909],{"stretchy":1587},[7880,7881,7882,7884],"msup",{},[155,7883,7835],{},[1080,7885,7886],{},"2",[58,7888,1300],{},[1080,7890,4352],{},[58,7892,3918],{"stretchy":1587},[62,7894,7895],{"encoding":64},"\\langle x^2+1 \\rangle",[39,7897,7899,7949],{"className":7898,"ariaHidden":70},[69],[39,7900,7902,7906,7909,7940,7943,7946],{"className":7901},[74],[39,7903],{"className":7904,"style":7905},[78],"height:1.0641em;vertical-align:-0.25em;",[39,7907,3909],{"className":7908},[1637],[39,7910,7912,7915],{"className":7911},[173],[39,7913,7835],{"className":7914},[173,174],[39,7916,7918],{"className":7917},[1715],[39,7919,7921],{"className":7920},[1719],[39,7922,7924],{"className":7923},[1724],[39,7925,7928],{"className":7926,"style":7927},[1728],"height:0.8141em;",[39,7929,7931,7934],{"style":7930},"top:-3.063em;margin-right:0.05em;",[39,7932],{"className":7933,"style":1737},[1736],[39,7935,7937],{"className":7936},[1741,1742,1743,1744],[39,7938,7886],{"className":7939},[173,1744],[39,7941],{"className":7942,"style":676},[247],[39,7944,1300],{"className":7945},[680],[39,7947],{"className":7948,"style":676},[247],[39,7950,7952,7955,7958],{"className":7951},[74],[39,7953],{"className":7954,"style":926},[78],[39,7956,4352],{"className":7957},[173],[39,7959,3918],{"className":7960},[1662],[7443,7962,7963,8009],{},[39,7964,7966,7984],{"className":7965},[42],[39,7967,7969],{"className":7968},[46],[48,7970,7971],{"xmlns":50},[52,7972,7973,7981],{},[55,7974,7975,7978],{},[58,7976,7977],{},"≅",[155,7979,7980],{"mathvariant":7462},"C",[62,7982,7983],{"encoding":64},"\\cong \\mathbb{C}",[39,7985,7987,8000],{"className":7986,"ariaHidden":70},[69],[39,7988,7990,7994,7997],{"className":7989},[74],[39,7991],{"className":7992,"style":7993},[78],"height:0.589em;",[39,7995,7977],{"className":7996},[83],[39,7998],{"className":7999,"style":248},[247],[39,8001,8003,8006],{"className":8002},[74],[39,8004],{"className":8005,"style":7476},[78],[39,8007,7980],{"className":8008},[173,7480],"（复数域）",[7443,8011,8012,8102],{},[39,8013,8015,8040],{"className":8014},[42],[39,8016,8018],{"className":8017},[46],[48,8019,8020],{"xmlns":50},[52,8021,8022,8038],{},[55,8023,8024,8026,8032,8034,8036],{},[58,8025,3909],{"stretchy":1587},[7880,8027,8028,8030],{},[155,8029,7835],{},[1080,8031,7886],{},[58,8033,1300],{},[1080,8035,4352],{},[58,8037,3918],{"stretchy":1587},[62,8039,7895],{"encoding":64},[39,8041,8043,8090],{"className":8042,"ariaHidden":70},[69],[39,8044,8046,8049,8052,8081,8084,8087],{"className":8045},[74],[39,8047],{"className":8048,"style":7905},[78],[39,8050,3909],{"className":8051},[1637],[39,8053,8055,8058],{"className":8054},[173],[39,8056,7835],{"className":8057},[173,174],[39,8059,8061],{"className":8060},[1715],[39,8062,8064],{"className":8063},[1719],[39,8065,8067],{"className":8066},[1724],[39,8068,8070],{"className":8069,"style":7927},[1728],[39,8071,8072,8075],{"style":7930},[39,8073],{"className":8074,"style":1737},[1736],[39,8076,8078],{"className":8077},[1741,1742,1743,1744],[39,8079,7886],{"className":8080},[173,1744],[39,8082],{"className":8083,"style":676},[247],[39,8085,1300],{"className":8086},[680],[39,8088],{"className":8089,"style":676},[247],[39,8091,8093,8096,8099],{"className":8092},[74],[39,8094],{"className":8095,"style":926},[78],[39,8097,4352],{"className":8098},[173],[39,8100,3918],{"className":8101},[1662]," 是极大理想",[7328,8104,8105,8152,8193,8239],{},[7443,8106,8107],{},[24,8108,7815,8109],{},[39,8110,8112,8131],{"className":8111},[42],[39,8113,8115],{"className":8114},[46],[48,8116,8117],{"xmlns":50},[52,8118,8119,8129],{},[55,8120,8121,8123,8125,8127],{},[155,8122,191],{"mathvariant":7462},[58,8124,7832],{"stretchy":1587},[155,8126,7835],{},[58,8128,7838],{"stretchy":1587},[62,8130,7841],{"encoding":64},[39,8132,8134],{"className":8133,"ariaHidden":70},[69],[39,8135,8137,8140,8143,8146,8149],{"className":8136},[74],[39,8138],{"className":8139,"style":926},[78],[39,8141,191],{"className":8142},[173,7480],[39,8144,7832],{"className":8145},[1637],[39,8147,7835],{"className":8148},[173,174],[39,8150,7838],{"className":8151},[1662],[7443,8153,8154],{},[39,8155,8157,8175],{"className":8156},[42],[39,8158,8160],{"className":8159},[46],[48,8161,8162],{"xmlns":50},[52,8163,8164,8172],{},[55,8165,8166,8168,8170],{},[58,8167,3909],{"stretchy":1587},[155,8169,7835],{},[58,8171,3918],{"stretchy":1587},[62,8173,8174],{"encoding":64},"\\langle x \\rangle",[39,8176,8178],{"className":8177,"ariaHidden":70},[69],[39,8179,8181,8184,8187,8190],{"className":8180},[74],[39,8182],{"className":8183,"style":926},[78],[39,8185,3909],{"className":8186},[1637],[39,8188,7835],{"className":8189},[173,174],[39,8191,3918],{"className":8192},[1662],[7443,8194,8195,8238],{},[39,8196,8198,8214],{"className":8197},[42],[39,8199,8201],{"className":8200},[46],[48,8202,8203],{"xmlns":50},[52,8204,8205,8211],{},[55,8206,8207,8209],{},[58,8208,7977],{},[155,8210,191],{"mathvariant":7462},[62,8212,8213],{"encoding":64},"\\cong \\mathbb{R}",[39,8215,8217,8229],{"className":8216,"ariaHidden":70},[69],[39,8218,8220,8223,8226],{"className":8219},[74],[39,8221],{"className":8222,"style":7993},[78],[39,8224,7977],{"className":8225},[83],[39,8227],{"className":8228,"style":248},[247],[39,8230,8232,8235],{"className":8231},[74],[39,8233],{"className":8234,"style":7476},[78],[39,8236,191],{"className":8237},[173,7480],"（实数域）",[7443,8240,8241,8102],{},[39,8242,8244,8261],{"className":8243},[42],[39,8245,8247],{"className":8246},[46],[48,8248,8249],{"xmlns":50},[52,8250,8251,8259],{},[55,8252,8253,8255,8257],{},[58,8254,3909],{"stretchy":1587},[155,8256,7835],{},[58,8258,3918],{"stretchy":1587},[62,8260,8174],{"encoding":64},[39,8262,8264],{"className":8263,"ariaHidden":70},[69],[39,8265,8267,8270,8273,8276],{"className":8266},[74],[39,8268],{"className":8269,"style":926},[78],[39,8271,3909],{"className":8272},[1637],[39,8274,7835],{"className":8275},[173,174],[39,8277,3918],{"className":8278},[1662],[128,8280],{},[135,8282,8283],{"id":8283},"重点回顾",[1048,8285,8286,8424],{},[31,8287,8288,8291,8292,8320,8321,8394,8395,8423],{},[24,8289,8290],{},"构造逆元的证明","（定理 3.5.2 的 ",[39,8293,8295,8308],{"className":8294},[42],[39,8296,8298],{"className":8297},[46],[48,8299,8300],{"xmlns":50},[52,8301,8302,8306],{},[55,8303,8304],{},[58,8305,962],{},[62,8307,965],{"encoding":64},[39,8309,8311],{"className":8310,"ariaHidden":70},[69],[39,8312,8314,8317],{"className":8313},[74],[39,8315],{"className":8316,"style":975},[78],[39,8318,962],{"className":8319},[83]," 部分）：利用 ",[39,8322,8324,8349],{"className":8323},[42],[39,8325,8327],{"className":8326},[46],[48,8328,8329],{"xmlns":50},[52,8330,8331,8347],{},[55,8332,8333,8335,8337,8339,8341,8343,8345],{},[1080,8334,4352],{},[58,8336,329],{},[58,8338,3909],{"stretchy":1587},[155,8340,2953],{},[58,8342,323],{"separator":70},[155,8344,320],{},[58,8346,3918],{"stretchy":1587},[62,8348,4367],{"encoding":64},[39,8350,8352,8370],{"className":8351,"ariaHidden":70},[69],[39,8353,8355,8358,8361,8364,8367],{"className":8354},[74],[39,8356],{"className":8357,"style":4377},[78],[39,8359,4352],{"className":8360},[173],[39,8362],{"className":8363,"style":248},[247],[39,8365,329],{"className":8366},[83],[39,8368],{"className":8369,"style":248},[247],[39,8371,8373,8376,8379,8382,8385,8388,8391],{"className":8372},[74],[39,8374],{"className":8375,"style":926},[78],[39,8377,3909],{"className":8378},[1637],[39,8380,2953],{"className":8381,"style":2968},[173,174],[39,8383,323],{"className":8384},[350],[39,8386],{"className":8387,"style":354},[247],[39,8389,320],{"className":8390},[173,174],[39,8392,3918],{"className":8393},[1662]," 强行把 ",[39,8396,8398,8411],{"className":8397},[42],[39,8399,8401],{"className":8400},[46],[48,8402,8403],{"xmlns":50},[52,8404,8405,8409],{},[55,8406,8407],{},[1080,8408,4352],{},[62,8410,4352],{"encoding":64},[39,8412,8414],{"className":8413,"ariaHidden":70},[69],[39,8415,8417,8420],{"className":8416},[74],[39,8418],{"className":8419,"style":1132},[78],[39,8421,4352],{"className":8422},[173]," 变出来，是本节最核心的技巧。",[31,8425,8426,8566,8567,8597],{},[24,8427,8428],{},[39,8429,8431,8471],{"className":8430},[42],[39,8432,8434],{"className":8433},[46],[48,8435,8436],{"xmlns":50},[52,8437,8438,8468],{},[55,8439,8440,8442,8444,8446,8448,8450,8452,8458,8460,8462,8464,8466],{},[155,8441,191],{"mathvariant":7462},[58,8443,7832],{"stretchy":1587},[155,8445,7835],{},[58,8447,7838],{"stretchy":1587},[155,8449,911],{"mathvariant":224},[58,8451,3909],{"stretchy":1587},[7880,8453,8454,8456],{},[155,8455,7835],{},[1080,8457,7886],{},[58,8459,1300],{},[1080,8461,4352],{},[58,8463,3918],{"stretchy":1587},[58,8465,7977],{},[155,8467,7980],{"mathvariant":7462},[62,8469,8470],{"encoding":64},"\\mathbb{R}[x]\u002F\\langle x^2+1 \\rangle \\cong \\mathbb{C}",[39,8472,8474,8536,8557],{"className":8473,"ariaHidden":70},[69],[39,8475,8477,8480,8483,8486,8489,8492,8495,8498,8527,8530,8533],{"className":8476},[74],[39,8478],{"className":8479,"style":7905},[78],[39,8481,191],{"className":8482},[173,7480],[39,8484,7832],{"className":8485},[1637],[39,8487,7835],{"className":8488},[173,174],[39,8490,7838],{"className":8491},[1662],[39,8493,911],{"className":8494},[173],[39,8496,3909],{"className":8497},[1637],[39,8499,8501,8504],{"className":8500},[173],[39,8502,7835],{"className":8503},[173,174],[39,8505,8507],{"className":8506},[1715],[39,8508,8510],{"className":8509},[1719],[39,8511,8513],{"className":8512},[1724],[39,8514,8516],{"className":8515,"style":7927},[1728],[39,8517,8518,8521],{"style":7930},[39,8519],{"className":8520,"style":1737},[1736],[39,8522,8524],{"className":8523},[1741,1742,1743,1744],[39,8525,7886],{"className":8526},[173,1744],[39,8528],{"className":8529,"style":676},[247],[39,8531,1300],{"className":8532},[680],[39,8534],{"className":8535,"style":676},[247],[39,8537,8539,8542,8545,8548,8551,8554],{"className":8538},[74],[39,8540],{"className":8541,"style":926},[78],[39,8543,4352],{"className":8544},[173],[39,8546,3918],{"className":8547},[1662],[39,8549],{"className":8550,"style":248},[247],[39,8552,7977],{"className":8553},[83],[39,8555],{"className":8556,"style":248},[247],[39,8558,8560,8563],{"className":8559},[74],[39,8561],{"className":8562,"style":7476},[78],[39,8564,7980],{"className":8565},[173,7480],"：用商环构造出虚数单位 ",[39,8568,8570,8584],{"className":8569},[42],[39,8571,8573],{"className":8572},[46],[48,8574,8575],{"xmlns":50},[52,8576,8577,8582],{},[55,8578,8579],{},[155,8580,8581],{},"i",[62,8583,8581],{"encoding":64},[39,8585,8587],{"className":8586,"ariaHidden":70},[69],[39,8588,8590,8594],{"className":8589},[74],[39,8591],{"className":8592,"style":8593},[78],"height:0.6595em;",[39,8595,8581],{"className":8596},[173,174]," 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