[{"data":1,"prerenderedAt":6445},["ShallowReactive",2],{"note:math\u002Fabstract_algebra\u002Fring":3,"site-search-catalogue":6444},{"id":4,"title":5,"body":6,"description":6421,"extension":6437,"meta":6438,"navigation":6439,"path":6440,"seo":6441,"stem":6442,"__hash__":6443},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fring.md","环的基本理论",{"type":7,"value":8,"toc":6420},"minimark",[9,12,17,21,77,80,84,87,92,291,1902,1906,1909,2411,2611,2613,2617,2620,2624,2888,2893,3505,3509,3512,4510,4512,4516,4519,5234,5238],[10,11,5],"h1",{"id":5},[13,14,16],"h2",{"id":15},"_1-开篇总结-executive-summary","1. 开篇总结 (Executive Summary)",[18,19,20],"p",{},"本课程的核心目标是将代数结构从单一运算（群）推广到双运算（环）。",[18,22,23,27,28,76],{},[24,25,26],"strong",{},"核心洞察："," 环（Ring）是对整数集合 ",[29,30,33,57],"span",{"className":31},[32],"katex",[29,34,37],{"className":35},[36],"katex-mathml",[38,39,41],"math",{"xmlns":40},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[42,43,44,52],"semantics",{},[45,46,47],"mrow",{},[48,49,51],"mi",{"mathvariant":50},"double-struck","Z",[53,54,56],"annotation",{"encoding":55},"application\u002Fx-tex","\\mathbb{Z}",[29,58,62],{"className":59,"ariaHidden":61},[60],"katex-html","true",[29,63,66,71],{"className":64},[65],"base",[29,67],{"className":68,"style":70},[69],"strut","height:0.6889em;",[29,72,51],{"className":73},[74,75],"mord","mathbb"," 的抽象化，旨在研究“加法”与“乘法”共存时的代数性质。相比于群，环增加了结构的复杂性（两个运算），但也因为分配律的存在，引入了更丰富的代数工具。课程的难点在于理解**零因子（Zero Divisors）**的存在如何破坏了初等代数中的“消去律”。",[78,79],"hr",{},[13,81,83],{"id":82},"_2-环的公理化定义体系-axiomatic-definitions","2. 环的公理化定义体系 (Axiomatic Definitions)",[18,85,86],{},"首先给出环的通用定义，随后通过增加公理（Axioms）不断缩小范围，构建了特殊的环类。",[88,89,91],"h3",{"id":90},"_21-环-ring-的基础定义","2.1 环 (Ring) 的基础定义",[18,93,94,95,127,128,175,176,216,217,286,287,290],{},"设 ",[29,96,98,112],{"className":97},[32],[29,99,101],{"className":100},[36],[38,102,103],{"xmlns":40},[42,104,105,110],{},[45,106,107],{},[48,108,109],{},"R",[53,111,109],{"encoding":55},[29,113,115],{"className":114,"ariaHidden":61},[60],[29,116,118,122],{"className":117},[65],[29,119],{"className":120,"style":121},[69],"height:0.6833em;",[29,123,109],{"className":124,"style":126},[74,125],"mathnormal","margin-right:0.0077em;"," 是一个非空集合，配备两个二元运算：加法 ",[29,129,131,154],{"className":130},[32],[29,132,134],{"className":133},[36],[38,135,136],{"xmlns":40},[42,137,138,151],{},[45,139,140,145,148],{},[141,142,144],"mo",{"stretchy":143},"false","(",[141,146,147],{},"+",[141,149,150],{"stretchy":143},")",[53,152,153],{"encoding":55},"(+)",[29,155,157],{"className":156,"ariaHidden":61},[60],[29,158,160,164,168,171],{"className":159},[65],[29,161],{"className":162,"style":163},[69],"height:1em;vertical-align:-0.25em;",[29,165,144],{"className":166},[167],"mopen",[29,169,147],{"className":170},[74],[29,172,150],{"className":173},[174],"mclose"," 和乘法 ",[29,177,179,198],{"className":178},[32],[29,180,182],{"className":181},[36],[38,183,184],{"xmlns":40},[42,185,186,195],{},[45,187,188,190,193],{},[141,189,144],{"stretchy":143},[141,191,192],{},"⋅",[141,194,150],{"stretchy":143},[53,196,197],{"encoding":55},"(\\cdot)",[29,199,201],{"className":200,"ariaHidden":61},[60],[29,202,204,207,210,213],{"className":203},[65],[29,205],{"className":206,"style":163},[69],[29,208,144],{"className":209},[167],[29,211,192],{"className":212},[74],[29,214,150],{"className":215},[174],"。若满足以下三组公理，则称 ",[29,218,220,247],{"className":219},[32],[29,221,223],{"className":222},[36],[38,224,225],{"xmlns":40},[42,226,227,244],{},[45,228,229,231,233,236,238,240,242],{},[141,230,144],{"stretchy":143},[48,232,109],{},[141,234,235],{"separator":61},",",[141,237,147],{},[141,239,235],{"separator":61},[141,241,192],{},[141,243,150],{"stretchy":143},[53,245,246],{"encoding":55},"(R, +, \\cdot)",[29,248,250],{"className":249,"ariaHidden":61},[60],[29,251,253,256,259,262,266,271,274,277,280,283],{"className":252},[65],[29,254],{"className":255,"style":163},[69],[29,257,144],{"className":258},[167],[29,260,109],{"className":261,"style":126},[74,125],[29,263,235],{"className":264},[265],"mpunct",[29,267],{"className":268,"style":270},[269],"mspace","margin-right:0.1667em;",[29,272,147],{"className":273},[74],[29,275,235],{"className":276},[265],[29,278],{"className":279,"style":270},[269],[29,281,192],{"className":282},[74],[29,284,150],{"className":285},[174]," 为一个",[24,288,289],{},"环","：",[292,293,294,1175,1546],"ol",{},[295,296,297,290,300,352,353,356,357],"li",{},[24,298,299],{},"加法群公理",[29,301,303,325],{"className":302},[32],[29,304,306],{"className":305},[36],[38,307,308],{"xmlns":40},[42,309,310,322],{},[45,311,312,314,316,318,320],{},[141,313,144],{"stretchy":143},[48,315,109],{},[141,317,235],{"separator":61},[141,319,147],{},[141,321,150],{"stretchy":143},[53,323,324],{"encoding":55},"(R, +)",[29,326,328],{"className":327,"ariaHidden":61},[60],[29,329,331,334,337,340,343,346,349],{"className":330},[65],[29,332],{"className":333,"style":163},[69],[29,335,144],{"className":336},[167],[29,338,109],{"className":339,"style":126},[74,125],[29,341,235],{"className":342},[265],[29,344],{"className":345,"style":270},[269],[29,347,147],{"className":348},[74],[29,350,150],{"className":351},[174]," 构成",[24,354,355],{},"阿贝尔群（Abelian Group）","。\n",[358,359,360,524,689,869,1075],"ul",{},[295,361,362,363],{},"封闭性：",[29,364,366,414],{"className":365},[32],[29,367,369],{"className":368},[36],[38,370,371],{"xmlns":40},[42,372,373,411],{},[45,374,375,379,382,384,387,390,392,396,399,401,403,405,407,409],{},[48,376,378],{"mathvariant":377},"normal","∀",[48,380,381],{},"a",[141,383,235],{"separator":61},[48,385,386],{},"b",[141,388,389],{},"∈",[48,391,109],{},[393,394,395],"mtext",{},"  ",[141,397,398],{},"⟹",[393,400,395],{},[48,402,381],{},[141,404,147],{},[48,406,386],{},[141,408,389],{},[48,410,109],{},[53,412,413],{"encoding":55},"\\forall a, b \\in R \\implies a+b \\in R",[29,415,417,450,475,496,515],{"className":416,"ariaHidden":61},[60],[29,418,420,424,427,430,433,436,439,443,447],{"className":419},[65],[29,421],{"className":422,"style":423},[69],"height:0.8889em;vertical-align:-0.1944em;",[29,425,378],{"className":426},[74],[29,428,381],{"className":429},[74,125],[29,431,235],{"className":432},[265],[29,434],{"className":435,"style":270},[269],[29,437,386],{"className":438},[74,125],[29,440],{"className":441,"style":442},[269],"margin-right:0.2778em;",[29,444,389],{"className":445},[446],"mrel",[29,448],{"className":449,"style":442},[269],[29,451,453,457,460,463,466,469,472],{"className":452},[65],[29,454],{"className":455,"style":456},[69],"height:0.7073em;vertical-align:-0.024em;",[29,458,109],{"className":459,"style":126},[74,125],[29,461],{"className":462,"style":442},[269],[29,464],{"className":465,"style":442},[269],[29,467,398],{"className":468},[446],[29,470],{"className":471,"style":442},[269],[29,473],{"className":474,"style":442},[269],[29,476,478,482,485,489,493],{"className":477},[65],[29,479],{"className":480,"style":481},[69],"height:0.6667em;vertical-align:-0.0833em;",[29,483,381],{"className":484},[74,125],[29,486],{"className":487,"style":488},[269],"margin-right:0.2222em;",[29,490,147],{"className":491},[492],"mbin",[29,494],{"className":495,"style":488},[269],[29,497,499,503,506,509,512],{"className":498},[65],[29,500],{"className":501,"style":502},[69],"height:0.7335em;vertical-align:-0.0391em;",[29,504,386],{"className":505},[74,125],[29,507],{"className":508,"style":442},[269],[29,510,389],{"className":511},[446],[29,513],{"className":514,"style":442},[269],[29,516,518,521],{"className":517},[65],[29,519],{"className":520,"style":121},[69],[29,522,109],{"className":523,"style":126},[74,125],[295,525,526,527],{},"结合律：",[29,528,530,574],{"className":529},[32],[29,531,533],{"className":532},[36],[38,534,535],{"xmlns":40},[42,536,537,571],{},[45,538,539,541,543,545,547,549,551,554,557,559,561,563,565,567,569],{},[141,540,144],{"stretchy":143},[48,542,381],{},[141,544,147],{},[48,546,386],{},[141,548,150],{"stretchy":143},[141,550,147],{},[48,552,553],{},"c",[141,555,556],{},"=",[48,558,381],{},[141,560,147],{},[141,562,144],{"stretchy":143},[48,564,386],{},[141,566,147],{},[48,568,553],{},[141,570,150],{"stretchy":143},[53,572,573],{"encoding":55},"(a+b)+c = a+(b+c)",[29,575,577,598,619,638,656,677],{"className":576,"ariaHidden":61},[60],[29,578,580,583,586,589,592,595],{"className":579},[65],[29,581],{"className":582,"style":163},[69],[29,584,144],{"className":585},[167],[29,587,381],{"className":588},[74,125],[29,590],{"className":591,"style":488},[269],[29,593,147],{"className":594},[492],[29,596],{"className":597,"style":488},[269],[29,599,601,604,607,610,613,616],{"className":600},[65],[29,602],{"className":603,"style":163},[69],[29,605,386],{"className":606},[74,125],[29,608,150],{"className":609},[174],[29,611],{"className":612,"style":488},[269],[29,614,147],{"className":615},[492],[29,617],{"className":618,"style":488},[269],[29,620,622,626,629,632,635],{"className":621},[65],[29,623],{"className":624,"style":625},[69],"height:0.4306em;",[29,627,553],{"className":628},[74,125],[29,630],{"className":631,"style":442},[269],[29,633,556],{"className":634},[446],[29,636],{"className":637,"style":442},[269],[29,639,641,644,647,650,653],{"className":640},[65],[29,642],{"className":643,"style":481},[69],[29,645,381],{"className":646},[74,125],[29,648],{"className":649,"style":488},[269],[29,651,147],{"className":652},[492],[29,654],{"className":655,"style":488},[269],[29,657,659,662,665,668,671,674],{"className":658},[65],[29,660],{"className":661,"style":163},[69],[29,663,144],{"className":664},[167],[29,666,386],{"className":667},[74,125],[29,669],{"className":670,"style":488},[269],[29,672,147],{"className":673},[492],[29,675],{"className":676,"style":488},[269],[29,678,680,683,686],{"className":679},[65],[29,681],{"className":682,"style":163},[69],[29,684,553],{"className":685},[74,125],[29,687,150],{"className":688},[174],[295,690,691,692,749,750],{},"单位元（零元）：",[29,693,695,718],{"className":694},[32],[29,696,698],{"className":697},[36],[38,699,700],{"xmlns":40},[42,701,702,715],{},[45,703,704,707,711,713],{},[48,705,706],{"mathvariant":377},"∃",[708,709,710],"mn",{},"0",[141,712,389],{},[48,714,109],{},[53,716,717],{"encoding":55},"\\exists 0 \\in R",[29,719,721,740],{"className":720,"ariaHidden":61},[60],[29,722,724,727,731,734,737],{"className":723},[65],[29,725],{"className":726,"style":502},[69],[29,728,730],{"className":729},[74],"∃0",[29,732],{"className":733,"style":442},[269],[29,735,389],{"className":736},[446],[29,738],{"className":739,"style":442},[269],[29,741,743,746],{"className":742},[65],[29,744],{"className":745,"style":121},[69],[29,747,109],{"className":748,"style":126},[74,125],"，使得 ",[29,751,753,783],{"className":752},[32],[29,754,756],{"className":755},[36],[38,757,758],{"xmlns":40},[42,759,760,780],{},[45,761,762,764,766,768,770,772,774,776,778],{},[48,763,381],{},[141,765,147],{},[708,767,710],{},[141,769,556],{},[708,771,710],{},[141,773,147],{},[48,775,381],{},[141,777,556],{},[48,779,381],{},[53,781,782],{"encoding":55},"a+0 = 0+a = a",[29,784,786,804,823,842,860],{"className":785,"ariaHidden":61},[60],[29,787,789,792,795,798,801],{"className":788},[65],[29,790],{"className":791,"style":481},[69],[29,793,381],{"className":794},[74,125],[29,796],{"className":797,"style":488},[269],[29,799,147],{"className":800},[492],[29,802],{"className":803,"style":488},[269],[29,805,807,811,814,817,820],{"className":806},[65],[29,808],{"className":809,"style":810},[69],"height:0.6444em;",[29,812,710],{"className":813},[74],[29,815],{"className":816,"style":442},[269],[29,818,556],{"className":819},[446],[29,821],{"className":822,"style":442},[269],[29,824,826,830,833,836,839],{"className":825},[65],[29,827],{"className":828,"style":829},[69],"height:0.7278em;vertical-align:-0.0833em;",[29,831,710],{"className":832},[74],[29,834],{"className":835,"style":488},[269],[29,837,147],{"className":838},[492],[29,840],{"className":841,"style":488},[269],[29,843,845,848,851,854,857],{"className":844},[65],[29,846],{"className":847,"style":625},[69],[29,849,381],{"className":850},[74,125],[29,852],{"className":853,"style":442},[269],[29,855,556],{"className":856},[446],[29,858],{"className":859,"style":442},[269],[29,861,863,866],{"className":862},[65],[29,864],{"className":865,"style":625},[69],[29,867,381],{"className":868},[74,125],[295,870,871,872,749,987],{},"逆元：",[29,873,875,908],{"className":874},[32],[29,876,878],{"className":877},[36],[38,879,880],{"xmlns":40},[42,881,882,905],{},[45,883,884,886,888,890,892,894,896,899,901,903],{},[48,885,378],{"mathvariant":377},[48,887,381],{},[141,889,389],{},[48,891,109],{},[141,893,235],{"separator":61},[48,895,706],{"mathvariant":377},[141,897,898],{},"−",[48,900,381],{},[141,902,389],{},[48,904,109],{},[53,906,907],{"encoding":55},"\\forall a \\in R, \\exists -a \\in R",[29,909,911,932,959,978],{"className":910,"ariaHidden":61},[60],[29,912,914,917,920,923,926,929],{"className":913},[65],[29,915],{"className":916,"style":502},[69],[29,918,378],{"className":919},[74],[29,921,381],{"className":922},[74,125],[29,924],{"className":925,"style":442},[269],[29,927,389],{"className":928},[446],[29,930],{"className":931,"style":442},[269],[29,933,935,938,941,944,947,950,953,956],{"className":934},[65],[29,936],{"className":937,"style":423},[69],[29,939,109],{"className":940,"style":126},[74,125],[29,942,235],{"className":943},[265],[29,945],{"className":946,"style":270},[269],[29,948,706],{"className":949},[74],[29,951],{"className":952,"style":488},[269],[29,954,898],{"className":955},[492],[29,957],{"className":958,"style":488},[269],[29,960,962,966,969,972,975],{"className":961},[65],[29,963],{"className":964,"style":965},[69],"height:0.5782em;vertical-align:-0.0391em;",[29,967,381],{"className":968},[74,125],[29,970],{"className":971,"style":442},[269],[29,973,389],{"className":974},[446],[29,976],{"className":977,"style":442},[269],[29,979,981,984],{"className":980},[65],[29,982],{"className":983,"style":121},[69],[29,985,109],{"className":986,"style":126},[74,125],[29,988,990,1018],{"className":989},[32],[29,991,993],{"className":992},[36],[38,994,995],{"xmlns":40},[42,996,997,1015],{},[45,998,999,1001,1003,1005,1007,1009,1011,1013],{},[48,1000,381],{},[141,1002,147],{},[141,1004,144],{"stretchy":143},[141,1006,898],{},[48,1008,381],{},[141,1010,150],{"stretchy":143},[141,1012,556],{},[708,1014,710],{},[53,1016,1017],{"encoding":55},"a+(-a) = 0",[29,1019,1021,1039,1066],{"className":1020,"ariaHidden":61},[60],[29,1022,1024,1027,1030,1033,1036],{"className":1023},[65],[29,1025],{"className":1026,"style":481},[69],[29,1028,381],{"className":1029},[74,125],[29,1031],{"className":1032,"style":488},[269],[29,1034,147],{"className":1035},[492],[29,1037],{"className":1038,"style":488},[269],[29,1040,1042,1045,1048,1051,1054,1057,1060,1063],{"className":1041},[65],[29,1043],{"className":1044,"style":163},[69],[29,1046,144],{"className":1047},[167],[29,1049,898],{"className":1050},[74],[29,1052,381],{"className":1053},[74,125],[29,1055,150],{"className":1056},[174],[29,1058],{"className":1059,"style":442},[269],[29,1061,556],{"className":1062},[446],[29,1064],{"className":1065,"style":442},[269],[29,1067,1069,1072],{"className":1068},[65],[29,1070],{"className":1071,"style":810},[69],[29,1073,710],{"className":1074},[74],[295,1076,1077,1078],{},"交换律：",[29,1079,1081,1107],{"className":1080},[32],[29,1082,1084],{"className":1083},[36],[38,1085,1086],{"xmlns":40},[42,1087,1088,1104],{},[45,1089,1090,1092,1094,1096,1098,1100,1102],{},[48,1091,381],{},[141,1093,147],{},[48,1095,386],{},[141,1097,556],{},[48,1099,386],{},[141,1101,147],{},[48,1103,381],{},[53,1105,1106],{"encoding":55},"a+b = b+a",[29,1108,1110,1128,1147,1166],{"className":1109,"ariaHidden":61},[60],[29,1111,1113,1116,1119,1122,1125],{"className":1112},[65],[29,1114],{"className":1115,"style":481},[69],[29,1117,381],{"className":1118},[74,125],[29,1120],{"className":1121,"style":488},[269],[29,1123,147],{"className":1124},[492],[29,1126],{"className":1127,"style":488},[269],[29,1129,1131,1135,1138,1141,1144],{"className":1130},[65],[29,1132],{"className":1133,"style":1134},[69],"height:0.6944em;",[29,1136,386],{"className":1137},[74,125],[29,1139],{"className":1140,"style":442},[269],[29,1142,556],{"className":1143},[446],[29,1145],{"className":1146,"style":442},[269],[29,1148,1150,1154,1157,1160,1163],{"className":1149},[65],[29,1151],{"className":1152,"style":1153},[69],"height:0.7778em;vertical-align:-0.0833em;",[29,1155,386],{"className":1156},[74,125],[29,1158],{"className":1159,"style":488},[269],[29,1161,147],{"className":1162},[492],[29,1164],{"className":1165,"style":488},[269],[29,1167,1169,1172],{"className":1168},[65],[29,1170],{"className":1171,"style":625},[69],[29,1173,381],{"className":1174},[74,125],[295,1176,1177,290,1180,352,1232,356,1235],{},[24,1178,1179],{},"乘法半群公理",[29,1181,1183,1205],{"className":1182},[32],[29,1184,1186],{"className":1185},[36],[38,1187,1188],{"xmlns":40},[42,1189,1190,1202],{},[45,1191,1192,1194,1196,1198,1200],{},[141,1193,144],{"stretchy":143},[48,1195,109],{},[141,1197,235],{"separator":61},[141,1199,192],{},[141,1201,150],{"stretchy":143},[53,1203,1204],{"encoding":55},"(R, \\cdot)",[29,1206,1208],{"className":1207,"ariaHidden":61},[60],[29,1209,1211,1214,1217,1220,1223,1226,1229],{"className":1210},[65],[29,1212],{"className":1213,"style":163},[69],[29,1215,144],{"className":1216},[167],[29,1218,109],{"className":1219,"style":126},[74,125],[29,1221,235],{"className":1222},[265],[29,1224],{"className":1225,"style":270},[269],[29,1227,192],{"className":1228},[74],[29,1230,150],{"className":1231},[174],[24,1233,1234],{},"半群（Semigroup）",[358,1236,1237,1385],{},[295,1238,362,1239],{},[29,1240,1242,1282],{"className":1241},[32],[29,1243,1245],{"className":1244},[36],[38,1246,1247],{"xmlns":40},[42,1248,1249,1279],{},[45,1250,1251,1253,1255,1257,1259,1261,1263,1265,1267,1269,1271,1273,1275,1277],{},[48,1252,378],{"mathvariant":377},[48,1254,381],{},[141,1256,235],{"separator":61},[48,1258,386],{},[141,1260,389],{},[48,1262,109],{},[393,1264,395],{},[141,1266,398],{},[393,1268,395],{},[48,1270,381],{},[141,1272,192],{},[48,1274,386],{},[141,1276,389],{},[48,1278,109],{},[53,1280,1281],{"encoding":55},"\\forall a, b \\in R \\implies a \\cdot b \\in R",[29,1283,1285,1315,1339,1358,1376],{"className":1284,"ariaHidden":61},[60],[29,1286,1288,1291,1294,1297,1300,1303,1306,1309,1312],{"className":1287},[65],[29,1289],{"className":1290,"style":423},[69],[29,1292,378],{"className":1293},[74],[29,1295,381],{"className":1296},[74,125],[29,1298,235],{"className":1299},[265],[29,1301],{"className":1302,"style":270},[269],[29,1304,386],{"className":1305},[74,125],[29,1307],{"className":1308,"style":442},[269],[29,1310,389],{"className":1311},[446],[29,1313],{"className":1314,"style":442},[269],[29,1316,1318,1321,1324,1327,1330,1333,1336],{"className":1317},[65],[29,1319],{"className":1320,"style":456},[69],[29,1322,109],{"className":1323,"style":126},[74,125],[29,1325],{"className":1326,"style":442},[269],[29,1328],{"className":1329,"style":442},[269],[29,1331,398],{"className":1332},[446],[29,1334],{"className":1335,"style":442},[269],[29,1337],{"className":1338,"style":442},[269],[29,1340,1342,1346,1349,1352,1355],{"className":1341},[65],[29,1343],{"className":1344,"style":1345},[69],"height:0.4445em;",[29,1347,381],{"className":1348},[74,125],[29,1350],{"className":1351,"style":488},[269],[29,1353,192],{"className":1354},[492],[29,1356],{"className":1357,"style":488},[269],[29,1359,1361,1364,1367,1370,1373],{"className":1360},[65],[29,1362],{"className":1363,"style":502},[69],[29,1365,386],{"className":1366},[74,125],[29,1368],{"className":1369,"style":442},[269],[29,1371,389],{"className":1372},[446],[29,1374],{"className":1375,"style":442},[269],[29,1377,1379,1382],{"className":1378},[65],[29,1380],{"className":1381,"style":121},[69],[29,1383,109],{"className":1384,"style":126},[74,125],[295,1386,526,1387],{},[29,1388,1390,1432],{"className":1389},[32],[29,1391,1393],{"className":1392},[36],[38,1394,1395],{"xmlns":40},[42,1396,1397,1429],{},[45,1398,1399,1401,1403,1405,1407,1409,1411,1413,1415,1417,1419,1421,1423,1425,1427],{},[141,1400,144],{"stretchy":143},[48,1402,381],{},[141,1404,192],{},[48,1406,386],{},[141,1408,150],{"stretchy":143},[141,1410,192],{},[48,1412,553],{},[141,1414,556],{},[48,1416,381],{},[141,1418,192],{},[141,1420,144],{"stretchy":143},[48,1422,386],{},[141,1424,192],{},[48,1426,553],{},[141,1428,150],{"stretchy":143},[53,1430,1431],{"encoding":55},"(a \\cdot b) \\cdot c = a \\cdot (b \\cdot c)",[29,1433,1435,1456,1477,1495,1513,1534],{"className":1434,"ariaHidden":61},[60],[29,1436,1438,1441,1444,1447,1450,1453],{"className":1437},[65],[29,1439],{"className":1440,"style":163},[69],[29,1442,144],{"className":1443},[167],[29,1445,381],{"className":1446},[74,125],[29,1448],{"className":1449,"style":488},[269],[29,1451,192],{"className":1452},[492],[29,1454],{"className":1455,"style":488},[269],[29,1457,1459,1462,1465,1468,1471,1474],{"className":1458},[65],[29,1460],{"className":1461,"style":163},[69],[29,1463,386],{"className":1464},[74,125],[29,1466,150],{"className":1467},[174],[29,1469],{"className":1470,"style":488},[269],[29,1472,192],{"className":1473},[492],[29,1475],{"className":1476,"style":488},[269],[29,1478,1480,1483,1486,1489,1492],{"className":1479},[65],[29,1481],{"className":1482,"style":625},[69],[29,1484,553],{"className":1485},[74,125],[29,1487],{"className":1488,"style":442},[269],[29,1490,556],{"className":1491},[446],[29,1493],{"className":1494,"style":442},[269],[29,1496,1498,1501,1504,1507,1510],{"className":1497},[65],[29,1499],{"className":1500,"style":1345},[69],[29,1502,381],{"className":1503},[74,125],[29,1505],{"className":1506,"style":488},[269],[29,1508,192],{"className":1509},[492],[29,1511],{"className":1512,"style":488},[269],[29,1514,1516,1519,1522,1525,1528,1531],{"className":1515},[65],[29,1517],{"className":1518,"style":163},[69],[29,1520,144],{"className":1521},[167],[29,1523,386],{"className":1524},[74,125],[29,1526],{"className":1527,"style":488},[269],[29,1529,192],{"className":1530},[492],[29,1532],{"className":1533,"style":488},[269],[29,1535,1537,1540,1543],{"className":1536},[65],[29,1538],{"className":1539,"style":163},[69],[29,1541,553],{"className":1542},[74,125],[29,1544,150],{"className":1545},[174],[295,1547,1548,1551,1552],{},[24,1549,1550],{},"相容性公理（分配律）","：乘法对加法满足分配律。\n",[358,1553,1554,1728],{},[295,1555,1556,1557],{},"左分配律：",[29,1558,1560,1602],{"className":1559},[32],[29,1561,1563],{"className":1562},[36],[38,1564,1565],{"xmlns":40},[42,1566,1567,1599],{},[45,1568,1569,1571,1573,1575,1577,1579,1581,1583,1585,1587,1589,1591,1593,1595,1597],{},[48,1570,381],{},[141,1572,192],{},[141,1574,144],{"stretchy":143},[48,1576,386],{},[141,1578,147],{},[48,1580,553],{},[141,1582,150],{"stretchy":143},[141,1584,556],{},[48,1586,381],{},[141,1588,192],{},[48,1590,386],{},[141,1592,147],{},[48,1594,381],{},[141,1596,192],{},[48,1598,553],{},[53,1600,1601],{"encoding":55},"a \\cdot (b+c) = a \\cdot b + a \\cdot c",[29,1603,1605,1623,1644,1665,1683,1701,1719],{"className":1604,"ariaHidden":61},[60],[29,1606,1608,1611,1614,1617,1620],{"className":1607},[65],[29,1609],{"className":1610,"style":1345},[69],[29,1612,381],{"className":1613},[74,125],[29,1615],{"className":1616,"style":488},[269],[29,1618,192],{"className":1619},[492],[29,1621],{"className":1622,"style":488},[269],[29,1624,1626,1629,1632,1635,1638,1641],{"className":1625},[65],[29,1627],{"className":1628,"style":163},[69],[29,1630,144],{"className":1631},[167],[29,1633,386],{"className":1634},[74,125],[29,1636],{"className":1637,"style":488},[269],[29,1639,147],{"className":1640},[492],[29,1642],{"className":1643,"style":488},[269],[29,1645,1647,1650,1653,1656,1659,1662],{"className":1646},[65],[29,1648],{"className":1649,"style":163},[69],[29,1651,553],{"className":1652},[74,125],[29,1654,150],{"className":1655},[174],[29,1657],{"className":1658,"style":442},[269],[29,1660,556],{"className":1661},[446],[29,1663],{"className":1664,"style":442},[269],[29,1666,1668,1671,1674,1677,1680],{"className":1667},[65],[29,1669],{"className":1670,"style":1345},[69],[29,1672,381],{"className":1673},[74,125],[29,1675],{"className":1676,"style":488},[269],[29,1678,192],{"className":1679},[492],[29,1681],{"className":1682,"style":488},[269],[29,1684,1686,1689,1692,1695,1698],{"className":1685},[65],[29,1687],{"className":1688,"style":1153},[69],[29,1690,386],{"className":1691},[74,125],[29,1693],{"className":1694,"style":488},[269],[29,1696,147],{"className":1697},[492],[29,1699],{"className":1700,"style":488},[269],[29,1702,1704,1707,1710,1713,1716],{"className":1703},[65],[29,1705],{"className":1706,"style":1345},[69],[29,1708,381],{"className":1709},[74,125],[29,1711],{"className":1712,"style":488},[269],[29,1714,192],{"className":1715},[492],[29,1717],{"className":1718,"style":488},[269],[29,1720,1722,1725],{"className":1721},[65],[29,1723],{"className":1724,"style":625},[69],[29,1726,553],{"className":1727},[74,125],[295,1729,1730,1731],{},"右分配律：",[29,1732,1734,1776],{"className":1733},[32],[29,1735,1737],{"className":1736},[36],[38,1738,1739],{"xmlns":40},[42,1740,1741,1773],{},[45,1742,1743,1745,1747,1749,1751,1753,1755,1757,1759,1761,1763,1765,1767,1769,1771],{},[141,1744,144],{"stretchy":143},[48,1746,381],{},[141,1748,147],{},[48,1750,386],{},[141,1752,150],{"stretchy":143},[141,1754,192],{},[48,1756,553],{},[141,1758,556],{},[48,1760,381],{},[141,1762,192],{},[48,1764,553],{},[141,1766,147],{},[48,1768,386],{},[141,1770,192],{},[48,1772,553],{},[53,1774,1775],{"encoding":55},"(a+b) \\cdot c = a \\cdot c + b \\cdot c",[29,1777,1779,1800,1821,1839,1857,1875,1893],{"className":1778,"ariaHidden":61},[60],[29,1780,1782,1785,1788,1791,1794,1797],{"className":1781},[65],[29,1783],{"className":1784,"style":163},[69],[29,1786,144],{"className":1787},[167],[29,1789,381],{"className":1790},[74,125],[29,1792],{"className":1793,"style":488},[269],[29,1795,147],{"className":1796},[492],[29,1798],{"className":1799,"style":488},[269],[29,1801,1803,1806,1809,1812,1815,1818],{"className":1802},[65],[29,1804],{"className":1805,"style":163},[69],[29,1807,386],{"className":1808},[74,125],[29,1810,150],{"className":1811},[174],[29,1813],{"className":1814,"style":488},[269],[29,1816,192],{"className":1817},[492],[29,1819],{"className":1820,"style":488},[269],[29,1822,1824,1827,1830,1833,1836],{"className":1823},[65],[29,1825],{"className":1826,"style":625},[69],[29,1828,553],{"className":1829},[74,125],[29,1831],{"className":1832,"style":442},[269],[29,1834,556],{"className":1835},[446],[29,1837],{"className":1838,"style":442},[269],[29,1840,1842,1845,1848,1851,1854],{"className":1841},[65],[29,1843],{"className":1844,"style":1345},[69],[29,1846,381],{"className":1847},[74,125],[29,1849],{"className":1850,"style":488},[269],[29,1852,192],{"className":1853},[492],[29,1855],{"className":1856,"style":488},[269],[29,1858,1860,1863,1866,1869,1872],{"className":1859},[65],[29,1861],{"className":1862,"style":481},[69],[29,1864,553],{"className":1865},[74,125],[29,1867],{"className":1868,"style":488},[269],[29,1870,147],{"className":1871},[492],[29,1873],{"className":1874,"style":488},[269],[29,1876,1878,1881,1884,1887,1890],{"className":1877},[65],[29,1879],{"className":1880,"style":1134},[69],[29,1882,386],{"className":1883},[74,125],[29,1885],{"className":1886,"style":488},[269],[29,1888,192],{"className":1889},[492],[29,1891],{"className":1892,"style":488},[269],[29,1894,1896,1899],{"className":1895},[65],[29,1897],{"className":1898,"style":625},[69],[29,1900,553],{"className":1901},[74,125],[88,1903,1905],{"id":1904},"_22-环的分类与层级结构","2.2 环的分类与层级结构",[18,1907,1908],{},"讲课中口语表达较为琐碎，现将其整理为严谨的层级框架：",[1910,1911,1912,1929],"table",{},[1913,1914,1915],"thead",{},[1916,1917,1918,1923,1926],"tr",{},[1919,1920,1922],"th",{"align":1921},"left","术语 (Term)",[1919,1924,1925],{"align":1921},"定义特征 (Defining Characteristics)",[1919,1927,1928],{"align":1921},"备注",[1930,1931,1932,1946,2134,2204,2305,2353],"tbody",{},[1916,1933,1934,1940,1943],{},[1935,1936,1937],"td",{"align":1921},[24,1938,1939],{},"环 (Ring)",[1935,1941,1942],{"align":1921},"基础定义（如上）",[1935,1944,1945],{"align":1921},"最一般的结构",[1916,1947,1948,1953,2038],{},[1935,1949,1950],{"align":1921},[24,1951,1952],{},"幺环 (Unital Ring)",[1935,1954,1955,1956,1985,1986,2037],{"align":1921},"乘法有单位元 ",[29,1957,1959,1973],{"className":1958},[32],[29,1960,1962],{"className":1961},[36],[38,1963,1964],{"xmlns":40},[42,1965,1966,1971],{},[45,1967,1968],{},[708,1969,1970],{},"1",[53,1972,1970],{"encoding":55},[29,1974,1976],{"className":1975,"ariaHidden":61},[60],[29,1977,1979,1982],{"className":1978},[65],[29,1980],{"className":1981,"style":810},[69],[29,1983,1970],{"className":1984},[74],"（即 ",[29,1987,1989,2010],{"className":1988},[32],[29,1990,1992],{"className":1991},[36],[38,1993,1994],{"xmlns":40},[42,1995,1996,2008],{},[45,1997,1998,2000,2002,2004,2006],{},[141,1999,144],{"stretchy":143},[48,2001,109],{},[141,2003,235],{"separator":61},[141,2005,192],{},[141,2007,150],{"stretchy":143},[53,2009,1204],{"encoding":55},[29,2011,2013],{"className":2012,"ariaHidden":61},[60],[29,2014,2016,2019,2022,2025,2028,2031,2034],{"className":2015},[65],[29,2017],{"className":2018,"style":163},[69],[29,2020,144],{"className":2021},[167],[29,2023,109],{"className":2024,"style":126},[74,125],[29,2026,235],{"className":2027},[265],[29,2029],{"className":2030,"style":270},[269],[29,2032,192],{"className":2033},[74],[29,2035,150],{"className":2036},[174]," 是幺半群）",[1935,2039,2040,2041,2133],{"align":1921},"要求 ",[29,2042,2044,2063],{"className":2043},[32],[29,2045,2047],{"className":2046},[36],[38,2048,2049],{"xmlns":40},[42,2050,2051,2060],{},[45,2052,2053,2055,2058],{},[708,2054,1970],{},[141,2056,2057],{"mathvariant":377},"≠",[708,2059,710],{},[53,2061,2062],{"encoding":55},"1 \\neq 0",[29,2064,2066,2124],{"className":2065,"ariaHidden":61},[60],[29,2067,2069,2072,2075,2078,2121],{"className":2068},[65],[29,2070],{"className":2071,"style":423},[69],[29,2073,1970],{"className":2074},[74],[29,2076],{"className":2077,"style":442},[269],[29,2079,2081,2114,2118],{"className":2080},[446],[29,2082,2084],{"className":2083},[446],[29,2085,2088],{"className":2086},[74,2087],"vbox",[29,2089,2092],{"className":2090},[2091],"thinbox",[29,2093,2096,2099,2110],{"className":2094},[2095],"rlap",[29,2097],{"className":2098,"style":423},[69],[29,2100,2103],{"className":2101},[2102],"inner",[29,2104,2106],{"className":2105},[74],[29,2107,2109],{"className":2108},[446],"",[29,2111],{"className":2112},[2113],"fix",[29,2115],{"className":2116},[269,2117],"nobreak",[29,2119,556],{"className":2120},[446],[29,2122],{"className":2123,"style":442},[269],[29,2125,2127,2130],{"className":2126},[65],[29,2128],{"className":2129,"style":810},[69],[29,2131,710],{"className":2132},[74]," (通常)",[1916,2135,2136,2141,2201],{},[1935,2137,2138],{"align":1921},[24,2139,2140],{},"交换环 (Commutative Ring)",[1935,2142,2143,2144,150],{"align":1921},"乘法满足交换律 (",[29,2145,2147,2169],{"className":2146},[32],[29,2148,2150],{"className":2149},[36],[38,2151,2152],{"xmlns":40},[42,2153,2154,2166],{},[45,2155,2156,2158,2160,2162,2164],{},[48,2157,381],{},[48,2159,386],{},[141,2161,556],{},[48,2163,386],{},[48,2165,381],{},[53,2167,2168],{"encoding":55},"ab=ba",[29,2170,2172,2191],{"className":2171,"ariaHidden":61},[60],[29,2173,2175,2178,2182,2185,2188],{"className":2174},[65],[29,2176],{"className":2177,"style":1134},[69],[29,2179,2181],{"className":2180},[74,125],"ab",[29,2183],{"className":2184,"style":442},[269],[29,2186,556],{"className":2187},[446],[29,2189],{"className":2190,"style":442},[269],[29,2192,2194,2197],{"className":2193},[65],[29,2195],{"className":2196,"style":1134},[69],[29,2198,2200],{"className":2199},[74,125],"ba",[1935,2202,2203],{"align":1921},"如多项式环",[1916,2205,2206,2211,2302],{},[1935,2207,2208],{"align":1921},[24,2209,2210],{},"除环 (Division Ring)",[1935,2212,2213,2214,2301],{"align":1921},"幺环，且非零元素均可逆（",[29,2215,2217,2250],{"className":2216},[32],[29,2218,2220],{"className":2219},[36],[38,2221,2222],{"xmlns":40},[42,2223,2224,2247],{},[45,2225,2226,2228,2230,2233,2236,2238,2241,2243,2245],{},[141,2227,144],{"stretchy":143},[48,2229,109],{},[141,2231,2232],{},"∖",[141,2234,2235],{"stretchy":143},"{",[708,2237,710],{},[141,2239,2240],{"stretchy":143},"}",[141,2242,235],{"separator":61},[141,2244,192],{},[141,2246,150],{"stretchy":143},[53,2248,2249],{"encoding":55},"(R\\setminus\\{0\\}, \\cdot)",[29,2251,2253,2274],{"className":2252,"ariaHidden":61},[60],[29,2254,2256,2259,2262,2265,2268,2271],{"className":2255},[65],[29,2257],{"className":2258,"style":163},[69],[29,2260,144],{"className":2261},[167],[29,2263,109],{"className":2264,"style":126},[74,125],[29,2266],{"className":2267,"style":488},[269],[29,2269,2232],{"className":2270},[492],[29,2272],{"className":2273,"style":488},[269],[29,2275,2277,2280,2283,2286,2289,2292,2295,2298],{"className":2276},[65],[29,2278],{"className":2279,"style":163},[69],[29,2281,2235],{"className":2282},[167],[29,2284,710],{"className":2285},[74],[29,2287,2240],{"className":2288},[174],[29,2290,235],{"className":2291},[265],[29,2293],{"className":2294,"style":270},[269],[29,2296,192],{"className":2297},[74],[29,2299,150],{"className":2300},[174]," 是群）",[1935,2303,2304],{"align":1921},"也称斜域 (Skew Field)",[1916,2306,2307,2312,2321],{},[1935,2308,2309],{"align":1921},[24,2310,2311],{},"整环 (Integral Domain)",[1935,2313,2314,2317,2318],{"align":1921},[24,2315,2316],{},"交换幺环","，且",[24,2319,2320],{},"无零因子",[1935,2322,2323,2324,2352],{"align":1921},"类似 ",[29,2325,2327,2340],{"className":2326},[32],[29,2328,2330],{"className":2329},[36],[38,2331,2332],{"xmlns":40},[42,2333,2334,2338],{},[45,2335,2336],{},[48,2337,51],{"mathvariant":50},[53,2339,56],{"encoding":55},[29,2341,2343],{"className":2342,"ariaHidden":61},[60],[29,2344,2346,2349],{"className":2345},[65],[29,2347],{"className":2348,"style":70},[69],[29,2350,51],{"className":2351},[74,75]," 的结构",[1916,2354,2355,2360,2365],{},[1935,2356,2357],{"align":1921},[24,2358,2359],{},"域 (Field)",[1935,2361,2362],{"align":1921},[24,2363,2364],{},"交换除环",[1935,2366,2323,2367,2352],{"align":1921},[29,2368,2370,2389],{"className":2369},[32],[29,2371,2373],{"className":2372},[36],[38,2374,2375],{"xmlns":40},[42,2376,2377,2386],{},[45,2378,2379,2382,2384],{},[48,2380,2381],{"mathvariant":50},"Q",[141,2383,235],{"separator":61},[48,2385,109],{"mathvariant":50},[53,2387,2388],{"encoding":55},"\\mathbb{Q}, \\mathbb{R}",[29,2390,2392],{"className":2391,"ariaHidden":61},[60],[29,2393,2395,2399,2402,2405,2408],{"className":2394},[65],[29,2396],{"className":2397,"style":2398},[69],"height:0.8833em;vertical-align:-0.1944em;",[29,2400,2381],{"className":2401},[74,75],[29,2403,235],{"className":2404},[265],[29,2406],{"className":2407,"style":270},[269],[29,2409,109],{"className":2410},[74,75],[2412,2413,2414,2420],"blockquote",{},[18,2415,2416,2419],{},[24,2417,2418],{},"关键辨析："," 讲稿中对于“单位元”和“单位”做了区分，这是代数中的易混点。",[358,2421,2422,2457,2605],{},[295,2423,2424,2427,2428,2456],{},[24,2425,2426],{},"单位元 (Identity\u002FUnity):"," 指乘法幺元 ",[29,2429,2431,2444],{"className":2430},[32],[29,2432,2434],{"className":2433},[36],[38,2435,2436],{"xmlns":40},[42,2437,2438,2442],{},[45,2439,2440],{},[708,2441,1970],{},[53,2443,1970],{"encoding":55},[29,2445,2447],{"className":2446,"ariaHidden":61},[60],[29,2448,2450,2453],{"className":2449},[65],[29,2451],{"className":2452,"style":810},[69],[29,2454,1970],{"className":2455},[74],"。",[295,2458,2459,2462,2463,2492,2493,2523,2524,2456],{},[24,2460,2461],{},"单位 (Unit\u002FInvertible Element):"," 指乘法可逆的元素 ",[29,2464,2466,2480],{"className":2465},[32],[29,2467,2469],{"className":2468},[36],[38,2470,2471],{"xmlns":40},[42,2472,2473,2478],{},[45,2474,2475],{},[48,2476,2477],{},"u",[53,2479,2477],{"encoding":55},[29,2481,2483],{"className":2482,"ariaHidden":61},[60],[29,2484,2486,2489],{"className":2485},[65],[29,2487],{"className":2488,"style":625},[69],[29,2490,2477],{"className":2491},[74,125],"，即存在 ",[29,2494,2496,2510],{"className":2495},[32],[29,2497,2499],{"className":2498},[36],[38,2500,2501],{"xmlns":40},[42,2502,2503,2508],{},[45,2504,2505],{},[48,2506,2507],{},"v",[53,2509,2507],{"encoding":55},[29,2511,2513],{"className":2512,"ariaHidden":61},[60],[29,2514,2516,2519],{"className":2515},[65],[29,2517],{"className":2518,"style":625},[69],[29,2520,2507],{"className":2521,"style":2522},[74,125],"margin-right:0.0359em;"," 使得 ",[29,2525,2527,2553],{"className":2526},[32],[29,2528,2530],{"className":2529},[36],[38,2531,2532],{"xmlns":40},[42,2533,2534,2550],{},[45,2535,2536,2538,2540,2542,2544,2546,2548],{},[48,2537,2477],{},[48,2539,2507],{},[141,2541,556],{},[48,2543,2507],{},[48,2545,2477],{},[141,2547,556],{},[708,2549,1970],{},[53,2551,2552],{"encoding":55},"uv=vu=1",[29,2554,2556,2575,2596],{"className":2555,"ariaHidden":61},[60],[29,2557,2559,2562,2566,2569,2572],{"className":2558},[65],[29,2560],{"className":2561,"style":625},[69],[29,2563,2565],{"className":2564,"style":2522},[74,125],"uv",[29,2567],{"className":2568,"style":442},[269],[29,2570,556],{"className":2571},[446],[29,2573],{"className":2574,"style":442},[269],[29,2576,2578,2581,2584,2587,2590,2593],{"className":2577},[65],[29,2579],{"className":2580,"style":625},[69],[29,2582,2507],{"className":2583,"style":2522},[74,125],[29,2585,2477],{"className":2586},[74,125],[29,2588],{"className":2589,"style":442},[269],[29,2591,556],{"className":2592},[446],[29,2594],{"className":2595,"style":442},[269],[29,2597,2599,2602],{"className":2598},[65],[29,2600],{"className":2601,"style":810},[69],[29,2603,1970],{"className":2604},[74],[295,2606,2607,2610],{},[24,2608,2609],{},"域的定义："," 域既是整环（无零因子），又是除环（非零元可逆）。",[78,2612],{},[13,2614,2616],{"id":2615},"_3-核心概念深入分析零因子与整环","3. 核心概念深入分析：零因子与整环",[18,2618,2619],{},"零因子是理解环论与初等算术差异的关键。",[88,2621,2623],{"id":2622},"_31-零因子-zero-divisors","3.1 零因子 (Zero Divisors)",[18,2625,2626,2629,2630,2658,2659,749,2756,2829,2830,2858,2859,2887],{},[24,2627,2628],{},"定义："," 在环 ",[29,2631,2633,2646],{"className":2632},[32],[29,2634,2636],{"className":2635},[36],[38,2637,2638],{"xmlns":40},[42,2639,2640,2644],{},[45,2641,2642],{},[48,2643,109],{},[53,2645,109],{"encoding":55},[29,2647,2649],{"className":2648,"ariaHidden":61},[60],[29,2650,2652,2655],{"className":2651},[65],[29,2653],{"className":2654,"style":121},[69],[29,2656,109],{"className":2657,"style":126},[74,125]," 中，若存在非零元素 ",[29,2660,2662,2684],{"className":2661},[32],[29,2663,2665],{"className":2664},[36],[38,2666,2667],{"xmlns":40},[42,2668,2669,2681],{},[45,2670,2671,2673,2675,2677,2679],{},[48,2672,381],{},[141,2674,235],{"separator":61},[48,2676,386],{},[141,2678,2057],{"mathvariant":377},[708,2680,710],{},[53,2682,2683],{"encoding":55},"a, b \\neq 0",[29,2685,2687,2747],{"className":2686,"ariaHidden":61},[60],[29,2688,2690,2693,2696,2699,2702,2705,2708,2744],{"className":2689},[65],[29,2691],{"className":2692,"style":423},[69],[29,2694,381],{"className":2695},[74,125],[29,2697,235],{"className":2698},[265],[29,2700],{"className":2701,"style":270},[269],[29,2703,386],{"className":2704},[74,125],[29,2706],{"className":2707,"style":442},[269],[29,2709,2711,2738,2741],{"className":2710},[446],[29,2712,2714],{"className":2713},[446],[29,2715,2717],{"className":2716},[74,2087],[29,2718,2720],{"className":2719},[2091],[29,2721,2723,2726,2735],{"className":2722},[2095],[29,2724],{"className":2725,"style":423},[69],[29,2727,2729],{"className":2728},[2102],[29,2730,2732],{"className":2731},[74],[29,2733,2109],{"className":2734},[446],[29,2736],{"className":2737},[2113],[29,2739],{"className":2740},[269,2117],[29,2742,556],{"className":2743},[446],[29,2745],{"className":2746,"style":442},[269],[29,2748,2750,2753],{"className":2749},[65],[29,2751],{"className":2752,"style":810},[69],[29,2754,710],{"className":2755},[74],[29,2757,2759,2781],{"className":2758},[32],[29,2760,2762],{"className":2761},[36],[38,2763,2764],{"xmlns":40},[42,2765,2766,2778],{},[45,2767,2768,2770,2772,2774,2776],{},[48,2769,381],{},[141,2771,192],{},[48,2773,386],{},[141,2775,556],{},[708,2777,710],{},[53,2779,2780],{"encoding":55},"a \\cdot b = 0",[29,2782,2784,2802,2820],{"className":2783,"ariaHidden":61},[60],[29,2785,2787,2790,2793,2796,2799],{"className":2786},[65],[29,2788],{"className":2789,"style":1345},[69],[29,2791,381],{"className":2792},[74,125],[29,2794],{"className":2795,"style":488},[269],[29,2797,192],{"className":2798},[492],[29,2800],{"className":2801,"style":488},[269],[29,2803,2805,2808,2811,2814,2817],{"className":2804},[65],[29,2806],{"className":2807,"style":1134},[69],[29,2809,386],{"className":2810},[74,125],[29,2812],{"className":2813,"style":442},[269],[29,2815,556],{"className":2816},[446],[29,2818],{"className":2819,"style":442},[269],[29,2821,2823,2826],{"className":2822},[65],[29,2824],{"className":2825,"style":810},[69],[29,2827,710],{"className":2828},[74],"，则称 ",[29,2831,2833,2846],{"className":2832},[32],[29,2834,2836],{"className":2835},[36],[38,2837,2838],{"xmlns":40},[42,2839,2840,2844],{},[45,2841,2842],{},[48,2843,381],{},[53,2845,381],{"encoding":55},[29,2847,2849],{"className":2848,"ariaHidden":61},[60],[29,2850,2852,2855],{"className":2851},[65],[29,2853],{"className":2854,"style":625},[69],[29,2856,381],{"className":2857},[74,125]," 为左零因子，",[29,2860,2862,2875],{"className":2861},[32],[29,2863,2865],{"className":2864},[36],[38,2866,2867],{"xmlns":40},[42,2868,2869,2873],{},[45,2870,2871],{},[48,2872,386],{},[53,2874,386],{"encoding":55},[29,2876,2878],{"className":2877,"ariaHidden":61},[60],[29,2879,2881,2884],{"className":2880},[65],[29,2882],{"className":2883,"style":1134},[69],[29,2885,386],{"className":2886},[74,125]," 为右零因子。",[18,2889,2890],{},[24,2891,2892],{},"分析与洞察：",[358,2894,2895,3391],{},[295,2896,2897,2900,2901,2959,2960,3044,3045,3096,3097,3125,3126,356,3129],{},[24,2898,2899],{},"消去律的失效："," 在小学算术中，若 ",[29,2902,2904,2926],{"className":2903},[32],[29,2905,2907],{"className":2906},[36],[38,2908,2909],{"xmlns":40},[42,2910,2911,2923],{},[45,2912,2913,2915,2917,2919,2921],{},[48,2914,381],{},[48,2916,386],{},[141,2918,556],{},[48,2920,381],{},[48,2922,553],{},[53,2924,2925],{"encoding":55},"ab=ac",[29,2927,2929,2947],{"className":2928,"ariaHidden":61},[60],[29,2930,2932,2935,2938,2941,2944],{"className":2931},[65],[29,2933],{"className":2934,"style":1134},[69],[29,2936,2181],{"className":2937},[74,125],[29,2939],{"className":2940,"style":442},[269],[29,2942,556],{"className":2943},[446],[29,2945],{"className":2946,"style":442},[269],[29,2948,2950,2953,2956],{"className":2949},[65],[29,2951],{"className":2952,"style":625},[69],[29,2954,381],{"className":2955},[74,125],[29,2957,553],{"className":2958},[74,125]," 且 ",[29,2961,2963,2981],{"className":2962},[32],[29,2964,2966],{"className":2965},[36],[38,2967,2968],{"xmlns":40},[42,2969,2970,2978],{},[45,2971,2972,2974,2976],{},[48,2973,381],{},[141,2975,2057],{"mathvariant":377},[708,2977,710],{},[53,2979,2980],{"encoding":55},"a \\neq 0",[29,2982,2984,3035],{"className":2983,"ariaHidden":61},[60],[29,2985,2987,2990,2993,2996,3032],{"className":2986},[65],[29,2988],{"className":2989,"style":423},[69],[29,2991,381],{"className":2992},[74,125],[29,2994],{"className":2995,"style":442},[269],[29,2997,2999,3026,3029],{"className":2998},[446],[29,3000,3002],{"className":3001},[446],[29,3003,3005],{"className":3004},[74,2087],[29,3006,3008],{"className":3007},[2091],[29,3009,3011,3014,3023],{"className":3010},[2095],[29,3012],{"className":3013,"style":423},[69],[29,3015,3017],{"className":3016},[2102],[29,3018,3020],{"className":3019},[74],[29,3021,2109],{"className":3022},[446],[29,3024],{"className":3025},[2113],[29,3027],{"className":3028},[269,2117],[29,3030,556],{"className":3031},[446],[29,3033],{"className":3034,"style":442},[269],[29,3036,3038,3041],{"className":3037},[65],[29,3039],{"className":3040,"style":810},[69],[29,3042,710],{"className":3043},[74],"，我们可以推出 ",[29,3046,3048,3066],{"className":3047},[32],[29,3049,3051],{"className":3050},[36],[38,3052,3053],{"xmlns":40},[42,3054,3055,3063],{},[45,3056,3057,3059,3061],{},[48,3058,386],{},[141,3060,556],{},[48,3062,553],{},[53,3064,3065],{"encoding":55},"b=c",[29,3067,3069,3087],{"className":3068,"ariaHidden":61},[60],[29,3070,3072,3075,3078,3081,3084],{"className":3071},[65],[29,3073],{"className":3074,"style":1134},[69],[29,3076,386],{"className":3077},[74,125],[29,3079],{"className":3080,"style":442},[269],[29,3082,556],{"className":3083},[446],[29,3085],{"className":3086,"style":442},[269],[29,3088,3090,3093],{"className":3089},[65],[29,3091],{"className":3092,"style":625},[69],[29,3094,553],{"className":3095},[74,125],"。但在环论中，如果 ",[29,3098,3100,3113],{"className":3099},[32],[29,3101,3103],{"className":3102},[36],[38,3104,3105],{"xmlns":40},[42,3106,3107,3111],{},[45,3108,3109],{},[48,3110,381],{},[53,3112,381],{"encoding":55},[29,3114,3116],{"className":3115,"ariaHidden":61},[60],[29,3117,3119,3122],{"className":3118},[65],[29,3120],{"className":3121,"style":625},[69],[29,3123,381],{"className":3124},[74,125]," 是零因子，",[24,3127,3128],{},"消去律失效",[358,3130,3131],{},[295,3132,3133,3137,3138,3288,3289,3317,3318,2456],{},[3134,3135,3136],"em",{},"推导："," ",[29,3139,3141,3185],{"className":3140},[32],[29,3142,3144],{"className":3143},[36],[38,3145,3146],{"xmlns":40},[42,3147,3148,3182],{},[45,3149,3150,3152,3154,3156,3158,3160,3162,3164,3166,3168,3170,3172,3174,3176,3178,3180],{},[48,3151,381],{},[48,3153,386],{},[141,3155,556],{},[48,3157,381],{},[48,3159,553],{},[393,3161,395],{},[141,3163,398],{},[393,3165,395],{},[48,3167,381],{},[141,3169,144],{"stretchy":143},[48,3171,386],{},[141,3173,898],{},[48,3175,553],{},[141,3177,150],{"stretchy":143},[141,3179,556],{},[708,3181,710],{},[53,3183,3184],{"encoding":55},"ab=ac \\implies a(b-c)=0",[29,3186,3188,3206,3234,3258,3279],{"className":3187,"ariaHidden":61},[60],[29,3189,3191,3194,3197,3200,3203],{"className":3190},[65],[29,3192],{"className":3193,"style":1134},[69],[29,3195,2181],{"className":3196},[74,125],[29,3198],{"className":3199,"style":442},[269],[29,3201,556],{"className":3202},[446],[29,3204],{"className":3205,"style":442},[269],[29,3207,3209,3213,3216,3219,3222,3225,3228,3231],{"className":3208},[65],[29,3210],{"className":3211,"style":3212},[69],"height:0.549em;vertical-align:-0.024em;",[29,3214,381],{"className":3215},[74,125],[29,3217,553],{"className":3218},[74,125],[29,3220],{"className":3221,"style":442},[269],[29,3223],{"className":3224,"style":442},[269],[29,3226,398],{"className":3227},[446],[29,3229],{"className":3230,"style":442},[269],[29,3232],{"className":3233,"style":442},[269],[29,3235,3237,3240,3243,3246,3249,3252,3255],{"className":3236},[65],[29,3238],{"className":3239,"style":163},[69],[29,3241,381],{"className":3242},[74,125],[29,3244,144],{"className":3245},[167],[29,3247,386],{"className":3248},[74,125],[29,3250],{"className":3251,"style":488},[269],[29,3253,898],{"className":3254},[492],[29,3256],{"className":3257,"style":488},[269],[29,3259,3261,3264,3267,3270,3273,3276],{"className":3260},[65],[29,3262],{"className":3263,"style":163},[69],[29,3265,553],{"className":3266},[74,125],[29,3268,150],{"className":3269},[174],[29,3271],{"className":3272,"style":442},[269],[29,3274,556],{"className":3275},[446],[29,3277],{"className":3278,"style":442},[269],[29,3280,3282,3285],{"className":3281},[65],[29,3283],{"className":3284,"style":810},[69],[29,3286,710],{"className":3287},[74],"。若 ",[29,3290,3292,3305],{"className":3291},[32],[29,3293,3295],{"className":3294},[36],[38,3296,3297],{"xmlns":40},[42,3298,3299,3303],{},[45,3300,3301],{},[48,3302,381],{},[53,3304,381],{"encoding":55},[29,3306,3308],{"className":3307,"ariaHidden":61},[60],[29,3309,3311,3314],{"className":3310},[65],[29,3312],{"className":3313,"style":625},[69],[29,3315,381],{"className":3316},[74,125]," 是零因子，不能断定 ",[29,3319,3321,3343],{"className":3320},[32],[29,3322,3324],{"className":3323},[36],[38,3325,3326],{"xmlns":40},[42,3327,3328,3340],{},[45,3329,3330,3332,3334,3336,3338],{},[48,3331,386],{},[141,3333,898],{},[48,3335,553],{},[141,3337,556],{},[708,3339,710],{},[53,3341,3342],{"encoding":55},"b-c=0",[29,3344,3346,3364,3382],{"className":3345,"ariaHidden":61},[60],[29,3347,3349,3352,3355,3358,3361],{"className":3348},[65],[29,3350],{"className":3351,"style":1153},[69],[29,3353,386],{"className":3354},[74,125],[29,3356],{"className":3357,"style":488},[269],[29,3359,898],{"className":3360},[492],[29,3362],{"className":3363,"style":488},[269],[29,3365,3367,3370,3373,3376,3379],{"className":3366},[65],[29,3368],{"className":3369,"style":625},[69],[29,3371,553],{"className":3372},[74,125],[29,3374],{"className":3375,"style":442},[269],[29,3377,556],{"className":3378},[446],[29,3380],{"className":3381,"style":442},[269],[29,3383,3385,3388],{"className":3384},[65],[29,3386],{"className":3387,"style":810},[69],[29,3389,710],{"className":3390},[74],[295,3392,3393,3396,3397,3425,3426,3500,3501,3504],{},[24,3394,3395],{},"平凡零因子："," 元素 ",[29,3398,3400,3413],{"className":3399},[32],[29,3401,3403],{"className":3402},[36],[38,3404,3405],{"xmlns":40},[42,3406,3407,3411],{},[45,3408,3409],{},[708,3410,710],{},[53,3412,710],{"encoding":55},[29,3414,3416],{"className":3415,"ariaHidden":61},[60],[29,3417,3419,3422],{"className":3418},[65],[29,3420],{"className":3421,"style":810},[69],[29,3423,710],{"className":3424},[74]," 总是满足 ",[29,3427,3429,3452],{"className":3428},[32],[29,3430,3432],{"className":3431},[36],[38,3433,3434],{"xmlns":40},[42,3435,3436,3449],{},[45,3437,3438,3440,3442,3445,3447],{},[708,3439,710],{},[141,3441,192],{},[48,3443,3444],{},"x",[141,3446,556],{},[708,3448,710],{},[53,3450,3451],{"encoding":55},"0 \\cdot x = 0",[29,3453,3455,3473,3491],{"className":3454,"ariaHidden":61},[60],[29,3456,3458,3461,3464,3467,3470],{"className":3457},[65],[29,3459],{"className":3460,"style":810},[69],[29,3462,710],{"className":3463},[74],[29,3465],{"className":3466,"style":488},[269],[29,3468,192],{"className":3469},[492],[29,3471],{"className":3472,"style":488},[269],[29,3474,3476,3479,3482,3485,3488],{"className":3475},[65],[29,3477],{"className":3478,"style":625},[69],[29,3480,3444],{"className":3481},[74,125],[29,3483],{"className":3484,"style":442},[269],[29,3486,556],{"className":3487},[446],[29,3489],{"className":3490,"style":442},[269],[29,3492,3494,3497],{"className":3493},[65],[29,3495],{"className":3496,"style":810},[69],[29,3498,710],{"className":3499},[74],"，被称为平凡零因子。",[24,3502,3503],{},"整环","的精髓在于它排除了“非平凡零因子”，从而挽救了消去律。",[88,3506,3508],{"id":3507},"_32-基础性质推导","3.2 基础性质推导",[18,3510,3511],{},"讲稿中提及了三个命题，利用公理化定义可以严谨证明：",[292,3513,3514,4129],{},[295,3515,3516,3592],{},[24,3517,3518,3591],{},[29,3519,3521,3543],{"className":3520},[32],[29,3522,3524],{"className":3523},[36],[38,3525,3526],{"xmlns":40},[42,3527,3528,3540],{},[45,3529,3530,3532,3534,3536,3538],{},[708,3531,710],{},[141,3533,192],{},[48,3535,381],{},[141,3537,556],{},[708,3539,710],{},[53,3541,3542],{"encoding":55},"0 \\cdot a = 0",[29,3544,3546,3564,3582],{"className":3545,"ariaHidden":61},[60],[29,3547,3549,3552,3555,3558,3561],{"className":3548},[65],[29,3550],{"className":3551,"style":810},[69],[29,3553,710],{"className":3554},[74],[29,3556],{"className":3557,"style":488},[269],[29,3559,192],{"className":3560},[492],[29,3562],{"className":3563,"style":488},[269],[29,3565,3567,3570,3573,3576,3579],{"className":3566},[65],[29,3568],{"className":3569,"style":625},[69],[29,3571,381],{"className":3572},[74,125],[29,3574],{"className":3575,"style":442},[269],[29,3577,556],{"className":3578},[446],[29,3580],{"className":3581,"style":442},[269],[29,3583,3585,3588],{"className":3584},[65],[29,3586],{"className":3587,"style":810},[69],[29,3589,710],{"className":3590},[74]," 的证明：",[29,3593,3596],{"className":3594},[3595],"katex-display",[29,3597,3599,3758],{"className":3598},[32],[29,3600,3602],{"className":3601},[36],[38,3603,3605],{"xmlns":40,"display":3604},"block",[42,3606,3607,3755],{},[3608,3609,3613,3667,3718],"mtable",{"rowspacing":3610,"columnalign":3611,"columnspacing":3612},"0.25em","right left right","0em 1em",[3614,3615,3616,3630,3654],"mtr",{},[3617,3618,3619],"mtd",{},[3620,3621,3622],"mstyle",{"scriptlevel":710,"displaystyle":61},[45,3623,3624,3626,3628],{},[708,3625,710],{},[141,3627,192],{},[48,3629,381],{},[3617,3631,3632],{},[3620,3633,3634],{"scriptlevel":710,"displaystyle":61},[45,3635,3636,3638,3640,3642,3644,3646,3648,3650,3652],{},[45,3637],{},[141,3639,556],{},[141,3641,144],{"stretchy":143},[708,3643,710],{},[141,3645,147],{},[708,3647,710],{},[141,3649,150],{"stretchy":143},[141,3651,192],{},[48,3653,381],{},[3617,3655,3656],{},[3620,3657,3658],{"scriptlevel":710,"displaystyle":61},[45,3659,3660,3662,3665],{},[141,3661,144],{"stretchy":143},[393,3663,3664],{},"加法单位元定义",[141,3666,150],{"stretchy":143},[3614,3668,3669,3681,3705],{},[3617,3670,3671],{},[3620,3672,3673],{"scriptlevel":710,"displaystyle":61},[45,3674,3675,3677,3679],{},[708,3676,710],{},[141,3678,192],{},[48,3680,381],{},[3617,3682,3683],{},[3620,3684,3685],{"scriptlevel":710,"displaystyle":61},[45,3686,3687,3689,3691,3693,3695,3697,3699,3701,3703],{},[45,3688],{},[141,3690,556],{},[708,3692,710],{},[141,3694,192],{},[48,3696,381],{},[141,3698,147],{},[708,3700,710],{},[141,3702,192],{},[48,3704,381],{},[3617,3706,3707],{},[3620,3708,3709],{"scriptlevel":710,"displaystyle":61},[45,3710,3711,3713,3716],{},[141,3712,144],{"stretchy":143},[393,3714,3715],{},"分配律",[141,3717,150],{"stretchy":143},[3614,3719,3720,3726,3742],{},[3617,3721,3722],{},[3620,3723,3724],{"scriptlevel":710,"displaystyle":61},[708,3725,710],{},[3617,3727,3728],{},[3620,3729,3730],{"scriptlevel":710,"displaystyle":61},[45,3731,3732,3734,3736,3738,3740],{},[45,3733],{},[141,3735,556],{},[708,3737,710],{},[141,3739,192],{},[48,3741,381],{},[3617,3743,3744],{},[3620,3745,3746],{"scriptlevel":710,"displaystyle":61},[45,3747,3748,3750,3753],{},[141,3749,144],{"stretchy":143},[393,3751,3752],{},"加法群消去律",[141,3754,150],{"stretchy":143},[53,3756,3757],{"encoding":55},"\\begin{aligned}\n0 \\cdot a &= (0+0) \\cdot a & (\\text{加法单位元定义}) \\\\\n0 \\cdot a &= 0 \\cdot a + 0 \\cdot a & (\\text{分配律}) \\\\\n0 &= 0 \\cdot a & (\\text{加法群消去律})\n\\end{aligned}",[29,3759,3761],{"className":3760,"ariaHidden":61},[60],[29,3762,3764,3768],{"className":3763},[65],[29,3765],{"className":3766,"style":3767},[69],"height:4.2em;vertical-align:-1.85em;",[29,3769,3771],{"className":3770},[74],[29,3772,3774,3868,4039,4044],{"className":3773},[3608],[29,3775,3778],{"className":3776},[3777],"col-align-r",[29,3779,3783,3859],{"className":3780},[3781,3782],"vlist-t","vlist-t2",[29,3784,3787,3854],{"className":3785},[3786],"vlist-r",[29,3788,3792,3818,3842],{"className":3789,"style":3791},[3790],"vlist","height:2.35em;",[29,3793,3795,3800],{"style":3794},"top:-4.51em;",[29,3796],{"className":3797,"style":3799},[3798],"pstrut","height:3em;",[29,3801,3803,3806,3809,3812,3815],{"className":3802},[74],[29,3804,710],{"className":3805},[74],[29,3807],{"className":3808,"style":488},[269],[29,3810,192],{"className":3811},[492],[29,3813],{"className":3814,"style":488},[269],[29,3816,381],{"className":3817},[74,125],[29,3819,3821,3824],{"style":3820},"top:-3.01em;",[29,3822],{"className":3823,"style":3799},[3798],[29,3825,3827,3830,3833,3836,3839],{"className":3826},[74],[29,3828,710],{"className":3829},[74],[29,3831],{"className":3832,"style":488},[269],[29,3834,192],{"className":3835},[492],[29,3837],{"className":3838,"style":488},[269],[29,3840,381],{"className":3841},[74,125],[29,3843,3845,3848],{"style":3844},"top:-1.51em;",[29,3846],{"className":3847,"style":3799},[3798],[29,3849,3851],{"className":3850},[74],[29,3852,710],{"className":3853},[74],[29,3855,3858],{"className":3856},[3857],"vlist-s","​",[29,3860,3862],{"className":3861},[3786],[29,3863,3866],{"className":3864,"style":3865},[3790],"height:1.85em;",[29,3867],{},[29,3869,3872],{"className":3870},[3871],"col-align-l",[29,3873,3875,4031],{"className":3874},[3781,3782],[29,3876,3878,4028],{"className":3877},[3786],[29,3879,3881,3934,3993],{"className":3880,"style":3791},[3790],[29,3882,3883,3886],{"style":3794},[29,3884],{"className":3885,"style":3799},[3798],[29,3887,3889,3892,3895,3898,3901,3904,3907,3910,3913,3916,3919,3922,3925,3928,3931],{"className":3888},[74],[29,3890],{"className":3891},[74],[29,3893],{"className":3894,"style":442},[269],[29,3896,556],{"className":3897},[446],[29,3899],{"className":3900,"style":442},[269],[29,3902,144],{"className":3903},[167],[29,3905,710],{"className":3906},[74],[29,3908],{"className":3909,"style":488},[269],[29,3911,147],{"className":3912},[492],[29,3914],{"className":3915,"style":488},[269],[29,3917,710],{"className":3918},[74],[29,3920,150],{"className":3921},[174],[29,3923],{"className":3924,"style":488},[269],[29,3926,192],{"className":3927},[492],[29,3929],{"className":3930,"style":488},[269],[29,3932,381],{"className":3933},[74,125],[29,3935,3936,3939],{"style":3820},[29,3937],{"className":3938,"style":3799},[3798],[29,3940,3942,3945,3948,3951,3954,3957,3960,3963,3966,3969,3972,3975,3978,3981,3984,3987,3990],{"className":3941},[74],[29,3943],{"className":3944},[74],[29,3946],{"className":3947,"style":442},[269],[29,3949,556],{"className":3950},[446],[29,3952],{"className":3953,"style":442},[269],[29,3955,710],{"className":3956},[74],[29,3958],{"className":3959,"style":488},[269],[29,3961,192],{"className":3962},[492],[29,3964],{"className":3965,"style":488},[269],[29,3967,381],{"className":3968},[74,125],[29,3970],{"className":3971,"style":488},[269],[29,3973,147],{"className":3974},[492],[29,3976],{"className":3977,"style":488},[269],[29,3979,710],{"className":3980},[74],[29,3982],{"className":3983,"style":488},[269],[29,3985,192],{"className":3986},[492],[29,3988],{"className":3989,"style":488},[269],[29,3991,381],{"className":3992},[74,125],[29,3994,3995,3998],{"style":3844},[29,3996],{"className":3997,"style":3799},[3798],[29,3999,4001,4004,4007,4010,4013,4016,4019,4022,4025],{"className":4000},[74],[29,4002],{"className":4003},[74],[29,4005],{"className":4006,"style":442},[269],[29,4008,556],{"className":4009},[446],[29,4011],{"className":4012,"style":442},[269],[29,4014,710],{"className":4015},[74],[29,4017],{"className":4018,"style":488},[269],[29,4020,192],{"className":4021},[492],[29,4023],{"className":4024,"style":488},[269],[29,4026,381],{"className":4027},[74,125],[29,4029,3858],{"className":4030},[3857],[29,4032,4034],{"className":4033},[3786],[29,4035,4037],{"className":4036,"style":3865},[3790],[29,4038],{},[29,4040],{"className":4041,"style":4043},[4042],"arraycolsep","width:1em;",[29,4045,4047],{"className":4046},[3777],[29,4048,4050,4121],{"className":4049},[3781,3782],[29,4051,4053,4118],{"className":4052},[3786],[29,4054,4056,4078,4098],{"className":4055,"style":3791},[3790],[29,4057,4058,4061],{"style":3794},[29,4059],{"className":4060,"style":3799},[3798],[29,4062,4064,4067,4075],{"className":4063},[74],[29,4065,144],{"className":4066},[167],[29,4068,4071],{"className":4069},[74,4070],"text",[29,4072,3664],{"className":4073},[74,4074],"cjk_fallback",[29,4076,150],{"className":4077},[174],[29,4079,4080,4083],{"style":3820},[29,4081],{"className":4082,"style":3799},[3798],[29,4084,4086,4089,4095],{"className":4085},[74],[29,4087,144],{"className":4088},[167],[29,4090,4092],{"className":4091},[74,4070],[29,4093,3715],{"className":4094},[74,4074],[29,4096,150],{"className":4097},[174],[29,4099,4100,4103],{"style":3844},[29,4101],{"className":4102,"style":3799},[3798],[29,4104,4106,4109,4115],{"className":4105},[74],[29,4107,144],{"className":4108},[167],[29,4110,4112],{"className":4111},[74,4070],[29,4113,3752],{"className":4114},[74,4074],[29,4116,150],{"className":4117},[174],[29,4119,3858],{"className":4120},[3857],[29,4122,4124],{"className":4123},[3786],[29,4125,4127],{"className":4126,"style":3865},[3790],[29,4128],{},[295,4130,4131,4222,4223,4311,4312,2456],{},[24,4132,4133,4134,290],{},"符号规则 ",[29,4135,4137,4171],{"className":4136},[32],[29,4138,4140],{"className":4139},[36],[38,4141,4142],{"xmlns":40},[42,4143,4144,4168],{},[45,4145,4146,4148,4150,4152,4154,4156,4158,4160,4162,4164,4166],{},[141,4147,144],{"stretchy":143},[141,4149,898],{},[48,4151,381],{},[141,4153,150],{"stretchy":143},[48,4155,386],{},[141,4157,556],{},[141,4159,898],{},[141,4161,144],{"stretchy":143},[48,4163,381],{},[48,4165,386],{},[141,4167,150],{"stretchy":143},[53,4169,4170],{"encoding":55},"(-a)b = -(ab)",[29,4172,4174,4204],{"className":4173,"ariaHidden":61},[60],[29,4175,4177,4180,4183,4186,4189,4192,4195,4198,4201],{"className":4176},[65],[29,4178],{"className":4179,"style":163},[69],[29,4181,144],{"className":4182},[167],[29,4184,898],{"className":4185},[74],[29,4187,381],{"className":4188},[74,125],[29,4190,150],{"className":4191},[174],[29,4193,386],{"className":4194},[74,125],[29,4196],{"className":4197,"style":442},[269],[29,4199,556],{"className":4200},[446],[29,4202],{"className":4203,"style":442},[269],[29,4205,4207,4210,4213,4216,4219],{"className":4206},[65],[29,4208],{"className":4209,"style":163},[69],[29,4211,898],{"className":4212},[74],[29,4214,144],{"className":4215},[167],[29,4217,2181],{"className":4218},[74,125],[29,4220,150],{"className":4221},[174],"\n这说明“负号”可以像我们直觉中那样在乘法中移动。它建立在 ",[29,4224,4226,4254],{"className":4225},[32],[29,4227,4229],{"className":4228},[36],[38,4230,4231],{"xmlns":40},[42,4232,4233,4251],{},[45,4234,4235,4237,4239,4241,4243,4245,4247,4249],{},[48,4236,381],{},[141,4238,147],{},[141,4240,144],{"stretchy":143},[141,4242,898],{},[48,4244,381],{},[141,4246,150],{"stretchy":143},[141,4248,556],{},[708,4250,710],{},[53,4252,4253],{"encoding":55},"a + (-a) = 0",[29,4255,4257,4275,4302],{"className":4256,"ariaHidden":61},[60],[29,4258,4260,4263,4266,4269,4272],{"className":4259},[65],[29,4261],{"className":4262,"style":481},[69],[29,4264,381],{"className":4265},[74,125],[29,4267],{"className":4268,"style":488},[269],[29,4270,147],{"className":4271},[492],[29,4273],{"className":4274,"style":488},[269],[29,4276,4278,4281,4284,4287,4290,4293,4296,4299],{"className":4277},[65],[29,4279],{"className":4280,"style":163},[69],[29,4282,144],{"className":4283},[167],[29,4285,898],{"className":4286},[74],[29,4288,381],{"className":4289},[74,125],[29,4291,150],{"className":4292},[174],[29,4294],{"className":4295,"style":442},[269],[29,4297,556],{"className":4298},[446],[29,4300],{"className":4301,"style":442},[269],[29,4303,4305,4308],{"className":4304},[65],[29,4306],{"className":4307,"style":810},[69],[29,4309,710],{"className":4310},[74]," 的基础上，通过分配律导出 ",[29,4313,4315,4369],{"className":4314},[32],[29,4316,4318],{"className":4317},[36],[38,4319,4320],{"xmlns":40},[42,4321,4322,4366],{},[45,4323,4324,4326,4328,4330,4332,4334,4336,4338,4340,4342,4344,4346,4348,4350,4352,4354,4356,4358,4360,4362,4364],{},[48,4325,381],{},[48,4327,386],{},[141,4329,147],{},[141,4331,144],{"stretchy":143},[141,4333,898],{},[48,4335,381],{},[141,4337,150],{"stretchy":143},[48,4339,386],{},[141,4341,556],{},[141,4343,144],{"stretchy":143},[48,4345,381],{},[141,4347,898],{},[48,4349,381],{},[141,4351,150],{"stretchy":143},[48,4353,386],{},[141,4355,556],{},[708,4357,710],{},[141,4359,192],{},[48,4361,386],{},[141,4363,556],{},[708,4365,710],{},[53,4367,4368],{"encoding":55},"ab + (-a)b = (a-a)b = 0 \\cdot b = 0",[29,4370,4372,4390,4420,4441,4465,4483,4501],{"className":4371,"ariaHidden":61},[60],[29,4373,4375,4378,4381,4384,4387],{"className":4374},[65],[29,4376],{"className":4377,"style":1153},[69],[29,4379,2181],{"className":4380},[74,125],[29,4382],{"className":4383,"style":488},[269],[29,4385,147],{"className":4386},[492],[29,4388],{"className":4389,"style":488},[269],[29,4391,4393,4396,4399,4402,4405,4408,4411,4414,4417],{"className":4392},[65],[29,4394],{"className":4395,"style":163},[69],[29,4397,144],{"className":4398},[167],[29,4400,898],{"className":4401},[74],[29,4403,381],{"className":4404},[74,125],[29,4406,150],{"className":4407},[174],[29,4409,386],{"className":4410},[74,125],[29,4412],{"className":4413,"style":442},[269],[29,4415,556],{"className":4416},[446],[29,4418],{"className":4419,"style":442},[269],[29,4421,4423,4426,4429,4432,4435,4438],{"className":4422},[65],[29,4424],{"className":4425,"style":163},[69],[29,4427,144],{"className":4428},[167],[29,4430,381],{"className":4431},[74,125],[29,4433],{"className":4434,"style":488},[269],[29,4436,898],{"className":4437},[492],[29,4439],{"className":4440,"style":488},[269],[29,4442,4444,4447,4450,4453,4456,4459,4462],{"className":4443},[65],[29,4445],{"className":4446,"style":163},[69],[29,4448,381],{"className":4449},[74,125],[29,4451,150],{"className":4452},[174],[29,4454,386],{"className":4455},[74,125],[29,4457],{"className":4458,"style":442},[269],[29,4460,556],{"className":4461},[446],[29,4463],{"className":4464,"style":442},[269],[29,4466,4468,4471,4474,4477,4480],{"className":4467},[65],[29,4469],{"className":4470,"style":810},[69],[29,4472,710],{"className":4473},[74],[29,4475],{"className":4476,"style":488},[269],[29,4478,192],{"className":4479},[492],[29,4481],{"className":4482,"style":488},[269],[29,4484,4486,4489,4492,4495,4498],{"className":4485},[65],[29,4487],{"className":4488,"style":1134},[69],[29,4490,386],{"className":4491},[74,125],[29,4493],{"className":4494,"style":442},[269],[29,4496,556],{"className":4497},[446],[29,4499],{"className":4500,"style":442},[269],[29,4502,4504,4507],{"className":4503},[65],[29,4505],{"className":4506,"style":810},[69],[29,4508,710],{"className":4509},[74],[78,4511],{},[13,4513,4515],{"id":4514},"_4-实例分析与反思-case-studies","4. 实例分析与反思 (Case Studies)",[18,4517,4518],{},"根据讲课内容，我对提及的例子进行了结构化对比分析，以展示不同环的特性。",[1910,4520,4521,4544],{},[1913,4522,4523],{},[1916,4524,4525,4528,4532,4535,4538,4541],{},[1919,4526,4527],{"align":1921},"例子",[1919,4529,4531],{"align":4530},"center","交换性?",[1919,4533,4534],{"align":4530},"含幺?",[1919,4536,4537],{"align":4530},"无零因子?",[1919,4539,4540],{"align":4530},"可逆性 (非零元)?",[1919,4542,4543],{"align":1921},"结论类型",[1930,4545,4546,4631,4682,4800,5039],{},[1916,4547,4548,4581,4584,4586,4588,4626],{},[1935,4549,4550],{"align":1921},[24,4551,4552,4553],{},"整数 ",[29,4554,4556,4569],{"className":4555},[32],[29,4557,4559],{"className":4558},[36],[38,4560,4561],{"xmlns":40},[42,4562,4563,4567],{},[45,4564,4565],{},[48,4566,51],{"mathvariant":50},[53,4568,56],{"encoding":55},[29,4570,4572],{"className":4571,"ariaHidden":61},[60],[29,4573,4575,4578],{"className":4574},[65],[29,4576],{"className":4577,"style":70},[69],[29,4579,51],{"className":4580},[74,75],[1935,4582,4583],{"align":4530},"是",[1935,4585,4583],{"align":4530},[1935,4587,4583],{"align":4530},[1935,4589,4590,4591,150],{"align":4530},"否 (仅 ",[29,4592,4594,4611],{"className":4593},[32],[29,4595,4597],{"className":4596},[36],[38,4598,4599],{"xmlns":40},[42,4600,4601,4608],{},[45,4602,4603,4606],{},[141,4604,4605],{},"±",[708,4607,1970],{},[53,4609,4610],{"encoding":55},"\\pm 1",[29,4612,4614],{"className":4613,"ariaHidden":61},[60],[29,4615,4617,4620,4623],{"className":4616},[65],[29,4618],{"className":4619,"style":829},[69],[29,4621,4605],{"className":4622},[74],[29,4624,1970],{"className":4625},[74],[1935,4627,4628,4630],{"align":1921},[24,4629,3503],{}," (Integral Domain)",[1916,4632,4633,4668,4670,4672,4674,4676],{},[1935,4634,4635],{"align":1921},[24,4636,4637,4638],{},"有理数 ",[29,4639,4641,4655],{"className":4640},[32],[29,4642,4644],{"className":4643},[36],[38,4645,4646],{"xmlns":40},[42,4647,4648,4652],{},[45,4649,4650],{},[48,4651,2381],{"mathvariant":50},[53,4653,4654],{"encoding":55},"\\mathbb{Q}",[29,4656,4658],{"className":4657,"ariaHidden":61},[60],[29,4659,4661,4665],{"className":4660},[65],[29,4662],{"className":4663,"style":4664},[69],"height:0.8556em;vertical-align:-0.1667em;",[29,4666,2381],{"className":4667},[74,75],[1935,4669,4583],{"align":4530},[1935,4671,4583],{"align":4530},[1935,4673,4583],{"align":4530},[1935,4675,4583],{"align":4530},[1935,4677,4678,4681],{"align":1921},[24,4679,4680],{},"域"," (Field)",[1916,4683,4684,4787,4790,4792,4794,4797],{},[1935,4685,4686],{"align":1921},[24,4687,4688,4689],{},"矩阵 ",[29,4690,4692,4719],{"className":4691},[32],[29,4693,4695],{"className":4694},[36],[38,4696,4697],{"xmlns":40},[42,4698,4699,4716],{},[45,4700,4701,4710,4712,4714],{},[4702,4703,4704,4707],"msub",{},[48,4705,4706],{},"M",[48,4708,4709],{},"n",[141,4711,144],{"stretchy":143},[48,4713,109],{"mathvariant":50},[141,4715,150],{"stretchy":143},[53,4717,4718],{"encoding":55},"M_n(\\mathbb{R})",[29,4720,4722],{"className":4721,"ariaHidden":61},[60],[29,4723,4725,4728,4778,4781,4784],{"className":4724},[65],[29,4726],{"className":4727,"style":163},[69],[29,4729,4731,4735],{"className":4730},[74],[29,4732,4706],{"className":4733,"style":4734},[74,125],"margin-right:0.109em;",[29,4736,4739],{"className":4737},[4738],"msupsub",[29,4740,4742,4769],{"className":4741},[3781,3782],[29,4743,4745,4766],{"className":4744},[3786],[29,4746,4749],{"className":4747,"style":4748},[3790],"height:0.1514em;",[29,4750,4752,4756],{"style":4751},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[29,4753],{"className":4754,"style":4755},[3798],"height:2.7em;",[29,4757,4763],{"className":4758},[4759,4760,4761,4762],"sizing","reset-size6","size3","mtight",[29,4764,4709],{"className":4765},[74,125,4762],[29,4767,3858],{"className":4768},[3857],[29,4770,4772],{"className":4771},[3786],[29,4773,4776],{"className":4774,"style":4775},[3790],"height:0.15em;",[29,4777],{},[29,4779,144],{"className":4780},[167],[29,4782,109],{"className":4783},[74,75],[29,4785,150],{"className":4786},[174],[1935,4788,4789],{"align":4530},"否",[1935,4791,4583],{"align":4530},[1935,4793,4789],{"align":4530},[1935,4795,4796],{"align":4530},"否 (仅非奇异矩阵)",[1935,4798,4799],{"align":1921},"非交换幺环 (有零因子)",[1916,4801,4802,4908,4910,4912,4943,4973],{},[1935,4803,4804],{"align":1921},[24,4805,4806,4807,4835,4836],{},"模 ",[29,4808,4810,4823],{"className":4809},[32],[29,4811,4813],{"className":4812},[36],[38,4814,4815],{"xmlns":40},[42,4816,4817,4821],{},[45,4818,4819],{},[48,4820,4709],{},[53,4822,4709],{"encoding":55},[29,4824,4826],{"className":4825,"ariaHidden":61},[60],[29,4827,4829,4832],{"className":4828},[65],[29,4830],{"className":4831,"style":625},[69],[29,4833,4709],{"className":4834},[74,125]," 整数 ",[29,4837,4839,4857],{"className":4838},[32],[29,4840,4842],{"className":4841},[36],[38,4843,4844],{"xmlns":40},[42,4845,4846,4854],{},[45,4847,4848],{},[4702,4849,4850,4852],{},[48,4851,51],{"mathvariant":50},[48,4853,4709],{},[53,4855,4856],{"encoding":55},"\\mathbb{Z}_n",[29,4858,4860],{"className":4859,"ariaHidden":61},[60],[29,4861,4863,4867],{"className":4862},[65],[29,4864],{"className":4865,"style":4866},[69],"height:0.8389em;vertical-align:-0.15em;",[29,4868,4870,4873],{"className":4869},[74],[29,4871,51],{"className":4872},[74,75],[29,4874,4876],{"className":4875},[4738],[29,4877,4879,4900],{"className":4878},[3781,3782],[29,4880,4882,4897],{"className":4881},[3786],[29,4883,4885],{"className":4884,"style":4748},[3790],[29,4886,4888,4891],{"style":4887},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[29,4889],{"className":4890,"style":4755},[3798],[29,4892,4894],{"className":4893},[4759,4760,4761,4762],[29,4895,4709],{"className":4896},[74,125,4762],[29,4898,3858],{"className":4899},[3857],[29,4901,4903],{"className":4902},[3786],[29,4904,4906],{"className":4905,"style":4775},[3790],[29,4907],{},[1935,4909,4583],{"align":4530},[1935,4911,4583],{"align":4530},[1935,4913,4914,4915],{"align":4530},"取决于 ",[29,4916,4918,4931],{"className":4917},[32],[29,4919,4921],{"className":4920},[36],[38,4922,4923],{"xmlns":40},[42,4924,4925,4929],{},[45,4926,4927],{},[48,4928,4709],{},[53,4930,4709],{"encoding":55},[29,4932,4934],{"className":4933,"ariaHidden":61},[60],[29,4935,4937,4940],{"className":4936},[65],[29,4938],{"className":4939,"style":625},[69],[29,4941,4709],{"className":4942},[74,125],[1935,4944,4914,4945],{"align":4530},[29,4946,4948,4961],{"className":4947},[32],[29,4949,4951],{"className":4950},[36],[38,4952,4953],{"xmlns":40},[42,4954,4955,4959],{},[45,4956,4957],{},[48,4958,4709],{},[53,4960,4709],{"encoding":55},[29,4962,4964],{"className":4963,"ariaHidden":61},[60],[29,4965,4967,4970],{"className":4966},[65],[29,4968],{"className":4969,"style":625},[69],[29,4971,4709],{"className":4972},[74,125],[1935,4974,4975,4976,5004,5005,4975,5008,5036,5037],{"align":1921},"若 ",[29,4977,4979,4992],{"className":4978},[32],[29,4980,4982],{"className":4981},[36],[38,4983,4984],{"xmlns":40},[42,4985,4986,4990],{},[45,4987,4988],{},[48,4989,4709],{},[53,4991,4709],{"encoding":55},[29,4993,4995],{"className":4994,"ariaHidden":61},[60],[29,4996,4998,5001],{"className":4997},[65],[29,4999],{"className":5000,"style":625},[69],[29,5002,4709],{"className":5003},[74,125]," 为合数，则有零因子；",[5006,5007],"br",{},[29,5009,5011,5024],{"className":5010},[32],[29,5012,5014],{"className":5013},[36],[38,5015,5016],{"xmlns":40},[42,5017,5018,5022],{},[45,5019,5020],{},[48,5021,4709],{},[53,5023,4709],{"encoding":55},[29,5025,5027],{"className":5026,"ariaHidden":61},[60],[29,5028,5030,5033],{"className":5029},[65],[29,5031],{"className":5032,"style":625},[69],[29,5034,4709],{"className":5035},[74,125]," 为素数，则是",[24,5038,4680],{},[1916,5040,5041,5092,5123,5125,5156,5159],{},[1935,5042,5043],{"align":1921},[24,5044,5045,5046],{},"多项式 ",[29,5047,5049,5071],{"className":5048},[32],[29,5050,5052],{"className":5051},[36],[38,5053,5054],{"xmlns":40},[42,5055,5056,5068],{},[45,5057,5058,5060,5063,5065],{},[48,5059,109],{},[141,5061,5062],{"stretchy":143},"[",[48,5064,3444],{},[141,5066,5067],{"stretchy":143},"]",[53,5069,5070],{"encoding":55},"R[x]",[29,5072,5074],{"className":5073,"ariaHidden":61},[60],[29,5075,5077,5080,5083,5086,5089],{"className":5076},[65],[29,5078],{"className":5079,"style":163},[69],[29,5081,109],{"className":5082,"style":126},[74,125],[29,5084,5062],{"className":5085},[167],[29,5087,3444],{"className":5088},[74,125],[29,5090,5067],{"className":5091},[174],[1935,5093,4975,5094,5122],{"align":4530},[29,5095,5097,5110],{"className":5096},[32],[29,5098,5100],{"className":5099},[36],[38,5101,5102],{"xmlns":40},[42,5103,5104,5108],{},[45,5105,5106],{},[48,5107,109],{},[53,5109,109],{"encoding":55},[29,5111,5113],{"className":5112,"ariaHidden":61},[60],[29,5114,5116,5119],{"className":5115},[65],[29,5117],{"className":5118,"style":121},[69],[29,5120,109],{"className":5121,"style":126},[74,125]," 交换",[1935,5124,4583],{"align":4530},[1935,5126,4975,5127,5155],{"align":4530},[29,5128,5130,5143],{"className":5129},[32],[29,5131,5133],{"className":5132},[36],[38,5134,5135],{"xmlns":40},[42,5136,5137,5141],{},[45,5138,5139],{},[48,5140,109],{},[53,5142,109],{"encoding":55},[29,5144,5146],{"className":5145,"ariaHidden":61},[60],[29,5147,5149,5152],{"className":5148},[65],[29,5150],{"className":5151,"style":121},[69],[29,5153,109],{"className":5154,"style":126},[74,125]," 无零因子",[1935,5157,5158],{"align":4530},"否 (仅常数项可逆)",[1935,5160,4975,5161,5189,5190,5233],{"align":1921},[29,5162,5164,5177],{"className":5163},[32],[29,5165,5167],{"className":5166},[36],[38,5168,5169],{"xmlns":40},[42,5170,5171,5175],{},[45,5172,5173],{},[48,5174,109],{},[53,5176,109],{"encoding":55},[29,5178,5180],{"className":5179,"ariaHidden":61},[60],[29,5181,5183,5186],{"className":5182},[65],[29,5184],{"className":5185,"style":121},[69],[29,5187,109],{"className":5188,"style":126},[74,125]," 是整环，则 ",[29,5191,5193,5212],{"className":5192},[32],[29,5194,5196],{"className":5195},[36],[38,5197,5198],{"xmlns":40},[42,5199,5200,5210],{},[45,5201,5202,5204,5206,5208],{},[48,5203,109],{},[141,5205,5062],{"stretchy":143},[48,5207,3444],{},[141,5209,5067],{"stretchy":143},[53,5211,5070],{"encoding":55},[29,5213,5215],{"className":5214,"ariaHidden":61},[60],[29,5216,5218,5221,5224,5227,5230],{"className":5217},[65],[29,5219],{"className":5220,"style":163},[69],[29,5222,109],{"className":5223,"style":126},[74,125],[29,5225,5062],{"className":5226},[167],[29,5228,3444],{"className":5229},[74,125],[29,5231,5067],{"className":5232},[174]," 是整环",[88,5235,5237],{"id":5236},"关键案例深度剖析","关键案例深度剖析：",[292,5239,5240,5869],{},[295,5241,5242,5331,5332],{},[24,5243,5244,5245,5330],{},"矩阵环 ",[29,5246,5248,5272],{"className":5247},[32],[29,5249,5251],{"className":5250},[36],[38,5252,5253],{"xmlns":40},[42,5254,5255,5269],{},[45,5256,5257,5263,5265,5267],{},[4702,5258,5259,5261],{},[48,5260,4706],{},[48,5262,4709],{},[141,5264,144],{"stretchy":143},[48,5266,109],{},[141,5268,150],{"stretchy":143},[53,5270,5271],{"encoding":55},"M_n(R)",[29,5273,5275],{"className":5274,"ariaHidden":61},[60],[29,5276,5278,5281,5321,5324,5327],{"className":5277},[65],[29,5279],{"className":5280,"style":163},[69],[29,5282,5284,5287],{"className":5283},[74],[29,5285,4706],{"className":5286,"style":4734},[74,125],[29,5288,5290],{"className":5289},[4738],[29,5291,5293,5313],{"className":5292},[3781,3782],[29,5294,5296,5310],{"className":5295},[3786],[29,5297,5299],{"className":5298,"style":4748},[3790],[29,5300,5301,5304],{"style":4751},[29,5302],{"className":5303,"style":4755},[3798],[29,5305,5307],{"className":5306},[4759,4760,4761,4762],[29,5308,4709],{"className":5309},[74,125,4762],[29,5311,3858],{"className":5312},[3857],[29,5314,5316],{"className":5315},[3786],[29,5317,5319],{"className":5318,"style":4775},[3790],[29,5320],{},[29,5322,144],{"className":5323},[167],[29,5325,109],{"className":5326,"style":126},[74,125],[29,5328,150],{"className":5329},[174]," 的非交换性与零因子：","\n这是最典型的“非良态”例子。\n",[358,5333,5334],{},[295,5335,5336,5337,5868],{},"举例：",[29,5338,5340,5467],{"className":5339},[32],[29,5341,5343],{"className":5342},[36],[38,5344,5345],{"xmlns":40},[42,5346,5347,5464],{},[45,5348,5349,5388,5390,5426,5428],{},[45,5350,5351,5353,5386],{},[141,5352,144],{"fence":61},[3608,5354,5358,5372],{"rowspacing":5355,"columnalign":5356,"columnspacing":5357},"0.16em","center center","1em",[3614,5359,5360,5366],{},[3617,5361,5362],{},[3620,5363,5364],{"scriptlevel":710,"displaystyle":143},[708,5365,710],{},[3617,5367,5368],{},[3620,5369,5370],{"scriptlevel":710,"displaystyle":143},[708,5371,1970],{},[3614,5373,5374,5380],{},[3617,5375,5376],{},[3620,5377,5378],{"scriptlevel":710,"displaystyle":143},[708,5379,710],{},[3617,5381,5382],{},[3620,5383,5384],{"scriptlevel":710,"displaystyle":143},[708,5385,710],{},[141,5387,150],{"fence":61},[141,5389,192],{},[45,5391,5392,5394,5424],{},[141,5393,144],{"fence":61},[3608,5395,5396,5410],{"rowspacing":5355,"columnalign":5356,"columnspacing":5357},[3614,5397,5398,5404],{},[3617,5399,5400],{},[3620,5401,5402],{"scriptlevel":710,"displaystyle":143},[708,5403,1970],{},[3617,5405,5406],{},[3620,5407,5408],{"scriptlevel":710,"displaystyle":143},[708,5409,710],{},[3614,5411,5412,5418],{},[3617,5413,5414],{},[3620,5415,5416],{"scriptlevel":710,"displaystyle":143},[708,5417,710],{},[3617,5419,5420],{},[3620,5421,5422],{"scriptlevel":710,"displaystyle":143},[708,5423,710],{},[141,5425,150],{"fence":61},[141,5427,556],{},[45,5429,5430,5432,5462],{},[141,5431,144],{"fence":61},[3608,5433,5434,5448],{"rowspacing":5355,"columnalign":5356,"columnspacing":5357},[3614,5435,5436,5442],{},[3617,5437,5438],{},[3620,5439,5440],{"scriptlevel":710,"displaystyle":143},[708,5441,710],{},[3617,5443,5444],{},[3620,5445,5446],{"scriptlevel":710,"displaystyle":143},[708,5447,710],{},[3614,5449,5450,5456],{},[3617,5451,5452],{},[3620,5453,5454],{"scriptlevel":710,"displaystyle":143},[708,5455,710],{},[3617,5457,5458],{},[3620,5459,5460],{"scriptlevel":710,"displaystyle":143},[708,5461,710],{},[141,5463,150],{"fence":61},[53,5465,5466],{"encoding":55},"\\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} = \\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix}",[29,5468,5470,5613,5745],{"className":5469,"ariaHidden":61},[60],[29,5471,5473,5477,5604,5607,5610],{"className":5472},[65],[29,5474],{"className":5475,"style":5476},[69],"height:2.4em;vertical-align:-0.95em;",[29,5478,5481,5490,5598],{"className":5479},[5480],"minner",[29,5482,5486],{"className":5483,"style":5485},[167,5484],"delimcenter","top:0em;",[29,5487,144],{"className":5488},[5489,4761],"delimsizing",[29,5491,5493],{"className":5492},[74],[29,5494,5496,5546,5550,5553],{"className":5495},[3608],[29,5497,5500],{"className":5498},[5499],"col-align-c",[29,5501,5503,5537],{"className":5502},[3781,3782],[29,5504,5506,5534],{"className":5505},[3786],[29,5507,5510,5522],{"className":5508,"style":5509},[3790],"height:1.45em;",[29,5511,5513,5516],{"style":5512},"top:-3.61em;",[29,5514],{"className":5515,"style":3799},[3798],[29,5517,5519],{"className":5518},[74],[29,5520,710],{"className":5521},[74],[29,5523,5525,5528],{"style":5524},"top:-2.41em;",[29,5526],{"className":5527,"style":3799},[3798],[29,5529,5531],{"className":5530},[74],[29,5532,710],{"className":5533},[74],[29,5535,3858],{"className":5536},[3857],[29,5538,5540],{"className":5539},[3786],[29,5541,5544],{"className":5542,"style":5543},[3790],"height:0.95em;",[29,5545],{},[29,5547],{"className":5548,"style":5549},[4042],"width:0.5em;",[29,5551],{"className":5552,"style":5549},[4042],[29,5554,5556],{"className":5555},[5499],[29,5557,5559,5590],{"className":5558},[3781,3782],[29,5560,5562,5587],{"className":5561},[3786],[29,5563,5565,5576],{"className":5564,"style":5509},[3790],[29,5566,5567,5570],{"style":5512},[29,5568],{"className":5569,"style":3799},[3798],[29,5571,5573],{"className":5572},[74],[29,5574,1970],{"className":5575},[74],[29,5577,5578,5581],{"style":5524},[29,5579],{"className":5580,"style":3799},[3798],[29,5582,5584],{"className":5583},[74],[29,5585,710],{"className":5586},[74],[29,5588,3858],{"className":5589},[3857],[29,5591,5593],{"className":5592},[3786],[29,5594,5596],{"className":5595,"style":5543},[3790],[29,5597],{},[29,5599,5601],{"className":5600,"style":5485},[174,5484],[29,5602,150],{"className":5603},[5489,4761],[29,5605],{"className":5606,"style":488},[269],[29,5608,192],{"className":5609},[492],[29,5611],{"className":5612,"style":488},[269],[29,5614,5616,5619,5736,5739,5742],{"className":5615},[65],[29,5617],{"className":5618,"style":5476},[69],[29,5620,5622,5628,5730],{"className":5621},[5480],[29,5623,5625],{"className":5624,"style":5485},[167,5484],[29,5626,144],{"className":5627},[5489,4761],[29,5629,5631],{"className":5630},[74],[29,5632,5634,5679,5682,5685],{"className":5633},[3608],[29,5635,5637],{"className":5636},[5499],[29,5638,5640,5671],{"className":5639},[3781,3782],[29,5641,5643,5668],{"className":5642},[3786],[29,5644,5646,5657],{"className":5645,"style":5509},[3790],[29,5647,5648,5651],{"style":5512},[29,5649],{"className":5650,"style":3799},[3798],[29,5652,5654],{"className":5653},[74],[29,5655,1970],{"className":5656},[74],[29,5658,5659,5662],{"style":5524},[29,5660],{"className":5661,"style":3799},[3798],[29,5663,5665],{"className":5664},[74],[29,5666,710],{"className":5667},[74],[29,5669,3858],{"className":5670},[3857],[29,5672,5674],{"className":5673},[3786],[29,5675,5677],{"className":5676,"style":5543},[3790],[29,5678],{},[29,5680],{"className":5681,"style":5549},[4042],[29,5683],{"className":5684,"style":5549},[4042],[29,5686,5688],{"className":5687},[5499],[29,5689,5691,5722],{"className":5690},[3781,3782],[29,5692,5694,5719],{"className":5693},[3786],[29,5695,5697,5708],{"className":5696,"style":5509},[3790],[29,5698,5699,5702],{"style":5512},[29,5700],{"className":5701,"style":3799},[3798],[29,5703,5705],{"className":5704},[74],[29,5706,710],{"className":5707},[74],[29,5709,5710,5713],{"style":5524},[29,5711],{"className":5712,"style":3799},[3798],[29,5714,5716],{"className":5715},[74],[29,5717,710],{"className":5718},[74],[29,5720,3858],{"className":5721},[3857],[29,5723,5725],{"className":5724},[3786],[29,5726,5728],{"className":5727,"style":5543},[3790],[29,5729],{},[29,5731,5733],{"className":5732,"style":5485},[174,5484],[29,5734,150],{"className":5735},[5489,4761],[29,5737],{"className":5738,"style":442},[269],[29,5740,556],{"className":5741},[446],[29,5743],{"className":5744,"style":442},[269],[29,5746,5748,5751],{"className":5747},[65],[29,5749],{"className":5750,"style":5476},[69],[29,5752,5754,5760,5862],{"className":5753},[5480],[29,5755,5757],{"className":5756,"style":5485},[167,5484],[29,5758,144],{"className":5759},[5489,4761],[29,5761,5763],{"className":5762},[74],[29,5764,5766,5811,5814,5817],{"className":5765},[3608],[29,5767,5769],{"className":5768},[5499],[29,5770,5772,5803],{"className":5771},[3781,3782],[29,5773,5775,5800],{"className":5774},[3786],[29,5776,5778,5789],{"className":5777,"style":5509},[3790],[29,5779,5780,5783],{"style":5512},[29,5781],{"className":5782,"style":3799},[3798],[29,5784,5786],{"className":5785},[74],[29,5787,710],{"className":5788},[74],[29,5790,5791,5794],{"style":5524},[29,5792],{"className":5793,"style":3799},[3798],[29,5795,5797],{"className":5796},[74],[29,5798,710],{"className":5799},[74],[29,5801,3858],{"className":5802},[3857],[29,5804,5806],{"className":5805},[3786],[29,5807,5809],{"className":5808,"style":5543},[3790],[29,5810],{},[29,5812],{"className":5813,"style":5549},[4042],[29,5815],{"className":5816,"style":5549},[4042],[29,5818,5820],{"className":5819},[5499],[29,5821,5823,5854],{"className":5822},[3781,3782],[29,5824,5826,5851],{"className":5825},[3786],[29,5827,5829,5840],{"className":5828,"style":5509},[3790],[29,5830,5831,5834],{"style":5512},[29,5832],{"className":5833,"style":3799},[3798],[29,5835,5837],{"className":5836},[74],[29,5838,710],{"className":5839},[74],[29,5841,5842,5845],{"style":5524},[29,5843],{"className":5844,"style":3799},[3798],[29,5846,5848],{"className":5847},[74],[29,5849,710],{"className":5850},[74],[29,5852,3858],{"className":5853},[3857],[29,5855,5857],{"className":5856},[3786],[29,5858,5860],{"className":5859,"style":5543},[3790],[29,5861],{},[29,5863,5865],{"className":5864,"style":5485},[174,5484],[29,5866,150],{"className":5867},[5489,4761],"。两个非零矩阵相乘为零，直观展示了零因子的存在。",[295,5870,5871,5943],{},[24,5872,5873,5942],{},[29,5874,5876,5893],{"className":5875},[32],[29,5877,5879],{"className":5878},[36],[38,5880,5881],{"xmlns":40},[42,5882,5883,5891],{},[45,5884,5885],{},[4702,5886,5887,5889],{},[48,5888,51],{"mathvariant":50},[48,5890,4709],{},[53,5892,4856],{"encoding":55},[29,5894,5896],{"className":5895,"ariaHidden":61},[60],[29,5897,5899,5902],{"className":5898},[65],[29,5900],{"className":5901,"style":4866},[69],[29,5903,5905,5908],{"className":5904},[74],[29,5906,51],{"className":5907},[74,75],[29,5909,5911],{"className":5910},[4738],[29,5912,5914,5934],{"className":5913},[3781,3782],[29,5915,5917,5931],{"className":5916},[3786],[29,5918,5920],{"className":5919,"style":4748},[3790],[29,5921,5922,5925],{"style":4887},[29,5923],{"className":5924,"style":4755},[3798],[29,5926,5928],{"className":5927},[4759,4760,4761,4762],[29,5929,4709],{"className":5930},[74,125,4762],[29,5932,3858],{"className":5933},[3857],[29,5935,5937],{"className":5936},[3786],[29,5938,5940],{"className":5939,"style":4775},[3790],[29,5941],{}," 的二象性：",[358,5944,5945,6315],{},[295,5946,5947,6019,6020,6252,6253,6314],{},[29,5948,5950,5969],{"className":5949},[32],[29,5951,5953],{"className":5952},[36],[38,5954,5955],{"xmlns":40},[42,5956,5957,5966],{},[45,5958,5959],{},[4702,5960,5961,5963],{},[48,5962,51],{"mathvariant":50},[708,5964,5965],{},"4",[53,5967,5968],{"encoding":55},"\\mathbb{Z}_4",[29,5970,5972],{"className":5971,"ariaHidden":61},[60],[29,5973,5975,5978],{"className":5974},[65],[29,5976],{"className":5977,"style":4866},[69],[29,5979,5981,5984],{"className":5980},[74],[29,5982,51],{"className":5983},[74,75],[29,5985,5987],{"className":5986},[4738],[29,5988,5990,6011],{"className":5989},[3781,3782],[29,5991,5993,6008],{"className":5992},[3786],[29,5994,5997],{"className":5995,"style":5996},[3790],"height:0.3011em;",[29,5998,5999,6002],{"style":4887},[29,6000],{"className":6001,"style":4755},[3798],[29,6003,6005],{"className":6004},[4759,4760,4761,4762],[29,6006,5965],{"className":6007},[74,4762],[29,6009,3858],{"className":6010},[3857],[29,6012,6014],{"className":6013},[3786],[29,6015,6017],{"className":6016,"style":4775},[3790],[29,6018],{}," 中，",[29,6021,6023,6068],{"className":6022},[32],[29,6024,6026],{"className":6025},[36],[38,6027,6028],{"xmlns":40},[42,6029,6030,6065],{},[45,6031,6032,6041,6043,6049,6051,6057,6059],{},[6033,6034,6035,6038],"mover",{"accent":61},[708,6036,6037],{},"2",[141,6039,6040],{},"ˉ",[141,6042,192],{},[6033,6044,6045,6047],{"accent":61},[708,6046,6037],{},[141,6048,6040],{},[141,6050,556],{},[6033,6052,6053,6055],{"accent":61},[708,6054,5965],{},[141,6056,6040],{},[141,6058,556],{},[6033,6060,6061,6063],{"accent":61},[708,6062,710],{},[141,6064,6040],{},[53,6066,6067],{"encoding":55},"\\bar{2} \\cdot \\bar{2} = \\bar{4} = \\bar{0}",[29,6069,6071,6123,6169,6215],{"className":6070,"ariaHidden":61},[60],[29,6072,6074,6078,6114,6117,6120],{"className":6073},[65],[29,6075],{"className":6076,"style":6077},[69],"height:0.7812em;",[29,6079,6082],{"className":6080},[74,6081],"accent",[29,6083,6085],{"className":6084},[3781],[29,6086,6088],{"className":6087},[3786],[29,6089,6091,6100],{"className":6090,"style":6077},[3790],[29,6092,6094,6097],{"style":6093},"top:-3em;",[29,6095],{"className":6096,"style":3799},[3798],[29,6098,6037],{"className":6099},[74],[29,6101,6103,6106],{"style":6102},"top:-3.2134em;",[29,6104],{"className":6105,"style":3799},[3798],[29,6107,6111],{"className":6108,"style":6110},[6109],"accent-body","left:-0.25em;",[29,6112,6040],{"className":6113},[74],[29,6115],{"className":6116,"style":488},[269],[29,6118,192],{"className":6119},[492],[29,6121],{"className":6122,"style":488},[269],[29,6124,6126,6129,6160,6163,6166],{"className":6125},[65],[29,6127],{"className":6128,"style":6077},[69],[29,6130,6132],{"className":6131},[74,6081],[29,6133,6135],{"className":6134},[3781],[29,6136,6138],{"className":6137},[3786],[29,6139,6141,6149],{"className":6140,"style":6077},[3790],[29,6142,6143,6146],{"style":6093},[29,6144],{"className":6145,"style":3799},[3798],[29,6147,6037],{"className":6148},[74],[29,6150,6151,6154],{"style":6102},[29,6152],{"className":6153,"style":3799},[3798],[29,6155,6157],{"className":6156,"style":6110},[6109],[29,6158,6040],{"className":6159},[74],[29,6161],{"className":6162,"style":442},[269],[29,6164,556],{"className":6165},[446],[29,6167],{"className":6168,"style":442},[269],[29,6170,6172,6175,6206,6209,6212],{"className":6171},[65],[29,6173],{"className":6174,"style":6077},[69],[29,6176,6178],{"className":6177},[74,6081],[29,6179,6181],{"className":6180},[3781],[29,6182,6184],{"className":6183},[3786],[29,6185,6187,6195],{"className":6186,"style":6077},[3790],[29,6188,6189,6192],{"style":6093},[29,6190],{"className":6191,"style":3799},[3798],[29,6193,5965],{"className":6194},[74],[29,6196,6197,6200],{"style":6102},[29,6198],{"className":6199,"style":3799},[3798],[29,6201,6203],{"className":6202,"style":6110},[6109],[29,6204,6040],{"className":6205},[74],[29,6207],{"className":6208,"style":442},[269],[29,6210,556],{"className":6211},[446],[29,6213],{"className":6214,"style":442},[269],[29,6216,6218,6221],{"className":6217},[65],[29,6219],{"className":6220,"style":6077},[69],[29,6222,6224],{"className":6223},[74,6081],[29,6225,6227],{"className":6226},[3781],[29,6228,6230],{"className":6229},[3786],[29,6231,6233,6241],{"className":6232,"style":6077},[3790],[29,6234,6235,6238],{"style":6093},[29,6236],{"className":6237,"style":3799},[3798],[29,6239,710],{"className":6240},[74],[29,6242,6243,6246],{"style":6102},[29,6244],{"className":6245,"style":3799},[3798],[29,6247,6249],{"className":6248,"style":6110},[6109],[29,6250,6040],{"className":6251},[74],"，所以 ",[29,6254,6256,6274],{"className":6255},[32],[29,6257,6259],{"className":6258},[36],[38,6260,6261],{"xmlns":40},[42,6262,6263,6271],{},[45,6264,6265],{},[6033,6266,6267,6269],{"accent":61},[708,6268,6037],{},[141,6270,6040],{},[53,6272,6273],{"encoding":55},"\\bar{2}",[29,6275,6277],{"className":6276,"ariaHidden":61},[60],[29,6278,6280,6283],{"className":6279},[65],[29,6281],{"className":6282,"style":6077},[69],[29,6284,6286],{"className":6285},[74,6081],[29,6287,6289],{"className":6288},[3781],[29,6290,6292],{"className":6291},[3786],[29,6293,6295,6303],{"className":6294,"style":6077},[3790],[29,6296,6297,6300],{"style":6093},[29,6298],{"className":6299,"style":3799},[3798],[29,6301,6037],{"className":6302},[74],[29,6304,6305,6308],{"style":6102},[29,6306],{"className":6307,"style":3799},[3798],[29,6309,6311],{"className":6310,"style":6110},[6109],[29,6312,6040],{"className":6313},[74]," 是零因子。",[295,6316,6317,6389,6390,6419],{},[29,6318,6320,6338],{"className":6319},[32],[29,6321,6323],{"className":6322},[36],[38,6324,6325],{"xmlns":40},[42,6326,6327,6335],{},[45,6328,6329],{},[4702,6330,6331,6333],{},[48,6332,51],{"mathvariant":50},[48,6334,18],{},[53,6336,6337],{"encoding":55},"\\mathbb{Z}_p",[29,6339,6341],{"className":6340,"ariaHidden":61},[60],[29,6342,6344,6348],{"className":6343},[65],[29,6345],{"className":6346,"style":6347},[69],"height:0.975em;vertical-align:-0.2861em;",[29,6349,6351,6354],{"className":6350},[74],[29,6352,51],{"className":6353},[74,75],[29,6355,6357],{"className":6356},[4738],[29,6358,6360,6380],{"className":6359},[3781,3782],[29,6361,6363,6377],{"className":6362},[3786],[29,6364,6366],{"className":6365,"style":4748},[3790],[29,6367,6368,6371],{"style":4887},[29,6369],{"className":6370,"style":4755},[3798],[29,6372,6374],{"className":6373},[4759,4760,4761,4762],[29,6375,18],{"className":6376},[74,125,4762],[29,6378,3858],{"className":6379},[3857],[29,6381,6383],{"className":6382},[3786],[29,6384,6387],{"className":6385,"style":6386},[3790],"height:0.2861em;",[29,6388],{}," 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