1050:
610:
1297:
is a square-zero extension. Thus, a nilpotent extension breaks up into successive square-zero extensions. Because of this, it is usually enough to study square-zero extensions in order to understand nilpotent extensions.
1292:
194:
710:
1016:
954:
488:
1178:
642:
767:
1367:
841:
874:
799:
301:
281:
130:
103:
1130:
894:
540:
1422:, Interscience Tracts in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers a division of John Wiley & Sons,
1348:
Typical references require sections be homomorphisms without elaborating whether 1 is preserved. But since we need to be able to identify
1189:
17:
142:
658:
237:
1460:
1427:
959:
903:
407:
1142:
615:
1313:
1061:
1486:
304:
734:
394:
1088:
of a ring (commutative or not) by an ideal whose square is zero. Such an extension is called a
820:
846:
1501:
1437:
772:
286:
266:
249:
133:
115:
88:
8:
1307:
1107:
204:
42:
1481:
1397:
1081:
879:
605:{\displaystyle \operatorname {Sym} (M)/\bigoplus _{n\geq 2}\operatorname {Sym} ^{n}(M)}
68:
1456:
1423:
1401:
1389:
645:
308:
110:
1415:
1379:
1030:
215:
49:
1476:
1450:
1433:
57:
35:
31:
1495:
1393:
1180:
of a
Noetherian commutative ring by the nilradical is a nilpotent extension.
75:
361:, and so two extensions are equivalent if there is a morphism between them.
1384:
358:
1049:
354:
1356:(see the trivial extension example), it seems 1 needs to be preserved.
1092:, a square extension or just an extension. For a square-zero ideal
319:
1487:
Extension of an associative algebra at
Encyclopedia of Mathematics
1021:
One interesting feature of this construction is that the module
1287:{\displaystyle 0\to I^{n}/I^{n-1}\to R/I^{n-1}\to R/I^{n}\to 0}
1135:
More generally, an extension by a nilpotent ideal is called a
189:{\displaystyle 0\to I\to E{\overset {\phi }{{}\to {}}}R\to 0.}
769:
is a section (note this section is a ring homomorphism since
1100:
is contained in the left and right annihilators of itself,
1316:, a statement about an extension by the Jacobson radical.
705:{\displaystyle 0\to M\to E{\overset {p}{{}\to {}}}R\to 0}
525:) yields the above formula; in particular we see that
1192:
1145:
1110:
962:
906:
882:
849:
823:
775:
737:
661:
618:
543:
410:
289:
269:
236:
is defined in the same way by replacing "ring" with "
145:
118:
91:
1286:
1172:
1124:
1010:
948:
888:
868:
835:
793:
761:
704:
636:
604:
482:
295:
275:
188:
124:
97:
27:Surjective ring homomorphism with a given codomain
1493:
397:of abelian groups. Define the multiplication on
1365:
1025:becomes an ideal of some new ring. In his book
312:
1011:{\displaystyle e\mapsto (\phi (e),e-\phi (e))}
949:{\displaystyle (E,\phi )\simeq (R\oplus I,p)}
483:{\displaystyle (a,x)\cdot (b,y)=(ab,ay+bx).}
364:
1366:Anderson, D. D.; Winders, M. (March 2009).
509:where ε squares to zero and expanding out (
1455:. Springer Science & Business Media.
1383:
1332:
1330:
1084:, it is common to consider an extension
1040:
652:. We then have the short exact sequence
1448:
1336:
1173:{\displaystyle R\to R_{\mathrm {red} }}
805:). Conversely, every trivial extension
637:{\displaystyle \operatorname {Sym} (M)}
30:For the ring-theoretic equivalent of a
14:
1494:
1414:
1327:
529:is a ring. It is sometimes called the
56:is the ring-theoretic equivalent of a
1044:
357:, such a morphism is necessarily an
24:
1470:
1449:Sernesi, Edoardo (20 April 2007).
1164:
1161:
1158:
801:is the multiplicative identity of
25:
1513:
1452:Deformations of Algebraic Schemes
313:§ Example: trivial extension
1048:
1482:infinitesimal extension at nLab
345:that induces the identities on
1408:
1372:Journal of Commutative Algebra
1359:
1342:
1278:
1257:
1230:
1196:
1149:
1005:
1002:
996:
981:
975:
969:
966:
943:
925:
919:
907:
788:
776:
762:{\displaystyle r\mapsto (r,0)}
756:
744:
741:
696:
682:
671:
665:
631:
625:
599:
593:
556:
550:
474:
447:
441:
429:
423:
411:
180:
166:
155:
149:
13:
1:
1320:
334:, is an algebra homomorphism
244:" and "abelian groups" with "
1314:Wedderburn principal theorem
1139:. For example, the quotient
7:
1301:
255:An extension is said to be
10:
1518:
1368:"Idealization of a Module"
719:is the projection. Hence,
373:be a commutative ring and
40:
29:
1477:algebra extension at nLab
1035:principle of idealization
900:using a section, we have
836:{\displaystyle R\oplus I}
365:Trivial extension example
18:Principle of idealization
41:Not to be confused with
1033:calls this process the
869:{\displaystyle I^{2}=0}
531:algebra of dual numbers
493:Note that identifying (
105:) consisting of a ring
36:Subring#Ring extensions
1385:10.1216/JCA-2009-1-1-3
1288:
1174:
1126:
1012:
950:
890:
876:. Indeed, identifying
870:
837:
795:
763:
731:. It is trivial since
706:
638:
606:
484:
322:between extensions of
297:
277:
190:
126:
99:
1289:
1175:
1127:
1090:square-zero extension
1041:Square-zero extension
1013:
951:
891:
871:
838:
796:
794:{\displaystyle (1,0)}
764:
707:
639:
607:
485:
298:
296:{\displaystyle \phi }
278:
276:{\displaystyle \phi }
191:
127:
125:{\displaystyle \phi }
100:
98:{\displaystyle \phi }
1190:
1143:
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960:
904:
880:
847:
821:
773:
735:
659:
616:
541:
408:
287:
267:
143:
134:short exact sequence
116:
89:
1308:Formally smooth map
1137:nilpotent extension
1125:{\displaystyle R/I}
723:is an extension of
136:of abelian groups:
132:that fits into the
43:Algebraic extension
1284:
1170:
1122:
1082:deformation theory
1060:. You can help by
1008:
946:
886:
866:
833:
791:
759:
702:
634:
602:
579:
537:can be defined as
480:
293:
273:
186:
122:
95:
1462:978-3-540-30615-3
1416:Nagata, Masayoshi
1078:
1077:
889:{\displaystyle R}
817:is isomorphic to
691:
646:symmetric algebra
564:
533:. Alternatively,
309:ring homomorphism
175:
111:ring homomorphism
54:algebra extension
16:(Redirected from
1509:
1466:
1441:
1440:
1412:
1406:
1405:
1387:
1363:
1357:
1352:as a subring of
1346:
1340:
1334:
1293:
1291:
1290:
1285:
1277:
1276:
1267:
1256:
1255:
1240:
1229:
1228:
1213:
1208:
1207:
1179:
1177:
1176:
1171:
1169:
1168:
1167:
1131:
1129:
1128:
1123:
1118:
1073:
1070:
1052:
1045:
1017:
1015:
1014:
1009:
955:
953:
952:
947:
896:as a subring of
895:
893:
892:
887:
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872:
867:
859:
858:
842:
840:
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611:
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299:
294:
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230:extension of an
216:commutative ring
203:isomorphic to a
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176:
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131:
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96:
50:abstract algebra
21:
1517:
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1512:
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1510:
1508:
1507:
1506:
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1491:
1473:
1471:Further reading
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1199:
1191:
1188:
1187:
1157:
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1152:
1144:
1141:
1140:
1114:
1109:
1106:
1105:
1074:
1068:
1065:
1058:needs expansion
1043:
961:
958:
957:
905:
902:
901:
881:
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877:
854:
850:
848:
845:
844:
822:
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774:
771:
770:
736:
733:
732:
685:
680:
679:
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660:
657:
656:
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614:
613:
584:
580:
568:
559:
542:
539:
538:
409:
406:
405:
367:
341:
288:
285:
284:
268:
265:
264:
205:two-sided ideal
169:
164:
163:
161:
144:
141:
140:
117:
114:
113:
90:
87:
86:
58:group extension
46:
39:
32:field extension
28:
23:
22:
15:
12:
11:
5:
1515:
1505:
1504:
1490:
1489:
1484:
1479:
1472:
1469:
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1467:
1461:
1443:
1442:
1428:
1407:
1358:
1341:
1325:
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1319:
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1317:
1310:
1303:
1300:
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1262:
1259:
1254:
1251:
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1244:
1239:
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1227:
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1217:
1212:
1206:
1202:
1198:
1195:
1166:
1163:
1160:
1155:
1151:
1148:
1121:
1117:
1113:
1080:Especially in
1076:
1075:
1055:
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1039:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
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936:
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930:
927:
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885:
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862:
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832:
829:
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746:
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667:
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621:
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583:
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571:
567:
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558:
555:
552:
549:
546:
491:
490:
479:
476:
473:
470:
467:
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461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
425:
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419:
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413:
366:
363:
292:
283:splits; i.e.,
272:
197:
196:
185:
182:
179:
174:
168:
160:
157:
154:
151:
148:
121:
94:
65:ring extension
26:
9:
6:
4:
3:
2:
1514:
1503:
1500:
1499:
1497:
1488:
1485:
1483:
1480:
1478:
1475:
1474:
1464:
1458:
1454:
1453:
1447:
1446:
1439:
1435:
1431:
1429:0-88275-228-6
1425:
1421:
1417:
1411:
1403:
1399:
1395:
1391:
1386:
1381:
1377:
1373:
1369:
1362:
1355:
1351:
1345:
1338:
1333:
1331:
1326:
1315:
1311:
1309:
1306:
1305:
1299:
1281:
1273:
1269:
1264:
1260:
1252:
1249:
1246:
1242:
1237:
1233:
1225:
1222:
1219:
1215:
1210:
1204:
1200:
1193:
1186:
1185:
1184:
1181:
1153:
1146:
1138:
1133:
1119:
1115:
1111:
1103:
1099:
1095:
1091:
1087:
1083:
1072:
1063:
1059:
1056:This section
1054:
1051:
1047:
1046:
1038:
1036:
1032:
1028:
1024:
1019:
999:
993:
990:
987:
984:
978:
972:
963:
940:
937:
934:
931:
928:
922:
916:
913:
910:
899:
883:
863:
860:
855:
851:
830:
827:
824:
816:
812:
808:
804:
785:
782:
779:
753:
750:
747:
738:
730:
726:
722:
718:
699:
693:
688:
674:
668:
662:
655:
654:
653:
651:
647:
628:
622:
619:
596:
590:
585:
581:
575:
572:
569:
565:
560:
553:
547:
544:
536:
532:
528:
524:
520:
516:
512:
508:
504:
500:
496:
477:
471:
468:
465:
462:
459:
456:
453:
450:
444:
438:
435:
432:
426:
420:
417:
414:
404:
403:
402:
400:
396:
392:
388:
384:
381:-module. Let
380:
376:
372:
362:
360:
356:
352:
348:
344:
337:
333:
329:
325:
321:
316:
314:
310:
306:
290:
270:
262:
258:
253:
251:
247:
243:
239:
235:
233:
227:
225:
220:
217:
212:
210:
206:
202:
183:
177:
172:
158:
152:
146:
139:
138:
137:
135:
119:
112:
108:
92:
84:
80:
77:
76:abelian group
73:
70:
66:
63:Precisely, a
61:
59:
55:
51:
44:
37:
33:
19:
1451:
1419:
1410:
1375:
1371:
1361:
1353:
1349:
1344:
1337:Sernesi 2007
1296:
1183:In general,
1182:
1136:
1134:
1101:
1097:
1093:
1089:
1085:
1079:
1066:
1062:adding to it
1057:
1034:
1026:
1022:
1020:
897:
814:
810:
806:
802:
728:
724:
720:
716:
714:
649:
534:
530:
526:
522:
518:
514:
510:
506:
502:
498:
494:
492:
398:
390:
386:
382:
378:
374:
370:
368:
350:
346:
339:
335:
331:
327:
323:
317:
260:
256:
254:
245:
241:
231:
229:
223:
222:
218:
213:
208:
200:
198:
106:
82:
78:
71:
64:
62:
53:
47:
1502:Ring theory
1420:Local Rings
1378:(1): 3–56.
1132:-bimodule.
1027:Local Rings
359:isomorphism
330:, over say
199:This makes
81:is a pair (
1321:References
1069:March 2023
395:direct sum
355:five lemma
307:that is a
226:-extension
1402:120720674
1394:1939-2346
1279:→
1258:→
1250:−
1231:→
1223:−
1197:→
1150:→
994:ϕ
991:−
973:ϕ
967:↦
932:⊕
923:≃
917:ϕ
828:⊕
742:↦
697:→
683:→
672:→
666:→
623:
591:
573:≥
566:⨁
548:
427:⋅
353:. By the
303:admits a
291:ϕ
271:ϕ
181:→
173:ϕ
167:→
156:→
150:→
120:ϕ
93:ϕ
1496:Category
1418:(1962),
1339:, 1.1.1.
1302:See also
1096:, since
320:morphism
234:-algebra
214:Given a
1438:0155856
644:is the
501:) with
393:be the
305:section
257:trivial
250:modules
238:algebra
1459:
1436:
1426:
1400:
1392:
1031:Nagata
715:where
612:where
259:or to
228:or an
109:and a
74:by an
34:, see
1398:S2CID
1104:is a
342:'
311:(see
261:split
240:over
221:, an
67:of a
52:, an
1457:ISBN
1424:ISBN
1390:ISSN
1312:The
956:via
369:Let
349:and
69:ring
1380:doi
1064:.
843:if
813:by
809:of
727:by
648:of
620:Sym
582:Sym
545:Sym
401:by
377:an
326:by
315:).
263:if
252:".
207:of
48:In
1498::
1434:MR
1432:,
1396:.
1388:.
1374:.
1370:.
1329:^
1037:.
1029:,
1018:.
523:εy
521:+
517:)(
515:εx
513:+
507:εx
505:+
497:,
389:⊕
385:=
338:→
318:A
211:.
184:0.
85:,
60:.
1465:.
1404:.
1382::
1376:1
1354:E
1350:R
1282:0
1274:n
1270:I
1265:/
1261:R
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1238:/
1234:R
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1112:R
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1071:)
1067:(
1023:M
1006:)
1003:)
1000:e
997:(
988:e
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979:e
976:(
970:(
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926:(
920:)
914:,
911:E
908:(
898:E
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861:=
856:2
852:I
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786:0
783:,
780:1
777:(
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748:r
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629:M
626:(
600:)
597:M
594:(
586:n
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570:n
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551:(
535:E
527:E
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503:a
499:x
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475:)
472:x
469:b
466:+
463:y
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430:(
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418:,
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391:M
387:R
383:E
379:R
375:M
371:R
351:R
347:I
340:E
336:E
332:A
328:I
324:R
248:-
246:A
242:A
232:A
224:A
219:A
209:E
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178:R
159:E
153:I
147:0
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83:E
79:I
72:R
45:.
38:.
20:)
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