1639:
851:
1275:
312:
745:
204:
104:
584:
1347:
498:
433:
369:
994:
954:, the map from the isomorphism classes of indecomposable injective modules to the spectrum is a bijection. Moreover, a complete set of representatives for those classes is given by
1412:
934:
1053:
1143:
1024:
670:
534:
1607:
1574:
1148:
774:
237:
19:
This article mainly concerns associated primes in general ring theory. For the specific usage in commutative ring theory, see also
1562:
691:
1660:
130:
159:
876:
on ideals (for example, any right or left
Noetherian ring) every nonzero module has at least one associated prime.
67:
1478:
873:
869:
550:
57:
1280:
886:
For a one-sided
Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable
458:
393:
329:
1454:(considered as a free module over itself), but its associated prime ideal (2) is not an associated prime of
957:
1352:
1670:
896:
1665:
1078:
20:
1638:
1033:
590:
while the rest of the associated primes (i.e., those properly containing associated primes) are called
35:
1093:
858:
149:
126:
1644:
1624:
1421:
891:
883:
has either zero or one associated primes, making uniform modules an example of coprimary modules.
42:
1603:
1570:
1063:
1544:
1487:
999:
887:
134:
27:
1617:
1584:
1613:
1599:
1580:
1566:
868:
It is possible, even for a commutative local ring, that the set of associated primes of a
629:
137:
647:
511:
1554:
1027:
880:
1654:
1591:
1425:
941:
685:
145:
206:
Also linked with the concept of "associated primes" of the ideal are the notions of
153:
49:
1476:
Picavet, Gabriel (1985). "Propriétés et applications de la notion de contenu".
443:. In commutative algebra the usual definition is different, but equivalent: if
1634:
1491:
632:
is coprimary if and only if it has exactly one associated prime. A submodule
118:(word play between the notation and the fact that an associated prime is an
21:
Primary decomposition § Primary decomposition from associated primes
1270:{\displaystyle I=((x^{2}+y^{2}+z^{2}+w^{2})\cdot (z^{3}-w^{3}-3x^{3}))}
846:{\displaystyle \mathrm {Ass} _{R}(M')\subseteq \mathrm {Ass} _{R}(M).}
1079:
Primary decomposition#Primary decomposition from associated primes
1069:
over any ring, there are only finitely many associated primes of
307:{\displaystyle \mathrm {Ann} _{R}(N)=\mathrm {Ann} _{R}(N')\,}
1598:, Graduate Texts in Mathematics No. 189, Berlin, New York:
747:; thus, the notion is a generalization of a primary ideal.
586:(with respect to the set-theoretic inclusion) are called
1077:
For the case for commutative
Noetherian rings, see also
1439:
a finite abelian group, then the associated primes of
755:
Most of these properties and assertions are given in (
64:. The set of associated primes is usually denoted by
1355:
1283:
1151:
1096:
1036:
1002:
960:
899:
777:
694:
650:
553:
514:
461:
396:
332:
240:
162:
70:
16:
Prime ideal that is an annihilator a prime submodule
1450:The group of order 2 is a quotient of the integers
740:{\displaystyle \operatorname {Ass} _{R}(R/I)=\{P\}}
1406:
1341:
1269:
1137:
1047:
1018:
988:
928:
845:
739:
664:
578:
528:
492:
427:
363:
306:
198:
98:
872:is empty. However, in any ring satisfying the
199:{\displaystyle \operatorname {Ass} _{R}(R/J).}
156:, and this set of prime ideals coincides with
1443:are exactly the primes dividing the order of
8:
734:
728:
99:{\displaystyle \operatorname {Ass} _{R}(M),}
579:{\displaystyle \operatorname {Ass} _{R}(M)}
1420:is the ring of integers, then non-trivial
144:is decomposed as a finite intersection of
1392:
1376:
1363:
1354:
1342:{\displaystyle (x^{2}+y^{2}+z^{2}+w^{2})}
1330:
1317:
1304:
1291:
1282:
1255:
1239:
1226:
1207:
1194:
1181:
1168:
1150:
1104:
1103:
1095:
1044:
1038:
1037:
1035:
1015:
1001:
985:
976:
975:
970:
959:
900:
898:
825:
814:
790:
779:
776:
714:
699:
693:
654:
649:
620: = 0 for some positive integer
558:
552:
518:
513:
489:
474:
463:
460:
424:
409:
398:
395:
360:
345:
334:
331:
303:
283:
272:
253:
242:
239:
182:
167:
161:
75:
69:
1468:
624:. A nonzero finitely generated module
493:{\displaystyle \mathrm {Ann} _{R}(m)\,}
428:{\displaystyle \mathrm {Ann} _{R}(N)\,}
364:{\displaystyle \mathrm {Ann} _{R}(N)\,}
989:{\displaystyle E(R/{\mathfrak {p}})\,}
129:, associated primes are linked to the
1407:{\displaystyle (z^{3}-w^{3}-3x^{3}).}
7:
950:: For a commutative Noetherian ring
944:, then this map becomes a bijection.
447:is commutative, an associated prime
131:Lasker–Noether primary decomposition
1565:, vol. 150, Berlin, New York:
1528:
1516:
1504:
1428:of prime power order are coprimary.
1039:
977:
929:{\displaystyle \mathrm {Spec} (R).}
865:, their associated primes coincide.
756:
910:
907:
904:
901:
821:
818:
815:
786:
783:
780:
470:
467:
464:
405:
402:
399:
341:
338:
335:
279:
276:
273:
249:
246:
243:
14:
1048:{\displaystyle {\mathfrak {p}}\,}
1637:
1055:ranges over the prime ideals of
605: = 0 for some nonzero
536:is isomorphic to a submodule of
1145:the associated prime ideals of
1398:
1356:
1336:
1284:
1264:
1261:
1219:
1213:
1161:
1158:
1138:{\displaystyle R=\mathbb {C} }
1132:
1108:
1012:
1006:
982:
964:
920:
914:
837:
831:
807:
796:
722:
708:
573:
567:
486:
480:
421:
415:
357:
351:
300:
289:
265:
259:
190:
176:
90:
84:
1:
1596:Lectures on modules and rings
1563:Graduate Texts in Mathematics
455:is a prime ideal of the form
1435:is the ring of integers and
152:of these primary ideals are
140:. Specifically, if an ideal
1687:
314:for any nonzero submodule
60:of a (prime) submodule of
18:
1492:10.1080/00927878508823275
1479:Communications in Algebra
874:ascending chain condition
870:finitely generated module
106:and sometimes called the
439:is a prime submodule of
390:is an ideal of the form
759:) starting on page 86.
1507:, p. 117, Ex 40B.
1408:
1343:
1271:
1139:
1049:
1020:
1019:{\displaystyle E(-)\,}
990:
930:
847:
741:
666:
580:
547:, minimal elements in
543:In a commutative ring
530:
500:for a nonzero element
494:
429:
365:
308:
200:
100:
1409:
1344:
1272:
1140:
1050:
1021:
991:
931:
848:
742:
667:
581:
531:
495:
430:
366:
322:. For a prime module
309:
201:
101:
1353:
1281:
1149:
1094:
1034:
1000:
958:
897:
775:
692:
648:
551:
512:
459:
394:
371:is a prime ideal in
330:
238:
160:
68:
1661:Commutative algebra
1629:Commutative algebra
1625:Matsumura, Hideyuki
1559:Commutative algebra
1549:Algèbre commutative
1422:free abelian groups
859:essential submodule
665:{\displaystyle M/N}
628:over a commutative
597:A module is called
529:{\displaystyle R/P}
234:if the annihilator
127:commutative algebra
1645:Mathematics portal
1404:
1339:
1267:
1135:
1045:
1016:
986:
926:
843:
737:
672:is coprimary with
662:
576:
526:
490:
425:
361:
304:
196:
96:
56:that arises as an
1609:978-0-387-98428-5
1576:978-0-387-94268-1
1486:(10): 2231–2265.
1064:Noetherian module
888:injective modules
1678:
1647:
1642:
1641:
1631:
1620:
1587:
1545:Nicolas Bourbaki
1532:
1526:
1520:
1514:
1508:
1502:
1496:
1495:
1473:
1424:and non-trivial
1413:
1411:
1410:
1405:
1397:
1396:
1381:
1380:
1368:
1367:
1348:
1346:
1345:
1340:
1335:
1334:
1322:
1321:
1309:
1308:
1296:
1295:
1276:
1274:
1273:
1268:
1260:
1259:
1244:
1243:
1231:
1230:
1212:
1211:
1199:
1198:
1186:
1185:
1173:
1172:
1144:
1142:
1141:
1136:
1107:
1054:
1052:
1051:
1046:
1043:
1042:
1025:
1023:
1022:
1017:
995:
993:
992:
987:
981:
980:
974:
935:
933:
932:
927:
913:
852:
850:
849:
844:
830:
829:
824:
806:
795:
794:
789:
746:
744:
743:
738:
718:
704:
703:
671:
669:
668:
663:
658:
585:
583:
582:
577:
563:
562:
535:
533:
532:
527:
522:
508:or equivalently
499:
497:
496:
491:
479:
478:
473:
434:
432:
431:
426:
414:
413:
408:
380:associated prime
370:
368:
367:
362:
350:
349:
344:
313:
311:
310:
305:
299:
288:
287:
282:
258:
257:
252:
205:
203:
202:
197:
186:
172:
171:
138:Noetherian rings
117:
105:
103:
102:
97:
80:
79:
32:associated prime
28:abstract algebra
1686:
1685:
1681:
1680:
1679:
1677:
1676:
1675:
1651:
1650:
1643:
1636:
1623:
1610:
1600:Springer-Verlag
1590:
1577:
1567:Springer-Verlag
1555:Eisenbud, David
1553:
1541:
1536:
1535:
1527:
1523:
1515:
1511:
1503:
1499:
1475:
1474:
1470:
1465:
1388:
1372:
1359:
1351:
1350:
1326:
1313:
1300:
1287:
1279:
1278:
1277:are the ideals
1251:
1235:
1222:
1203:
1190:
1177:
1164:
1147:
1146:
1092:
1091:
1087:
1032:
1031:
998:
997:
956:
955:
948:Matlis' Theorem
895:
894:
853:If in addition
813:
799:
778:
773:
772:
753:
695:
690:
689:
688:if and only if
646:
645:
630:Noetherian ring
592:embedded primes
588:isolated primes
554:
549:
548:
510:
509:
462:
457:
456:
397:
392:
391:
333:
328:
327:
292:
271:
241:
236:
235:
220:
212:embedded primes
208:isolated primes
163:
158:
157:
115:
71:
66:
65:
24:
17:
12:
11:
5:
1684:
1682:
1674:
1673:
1668:
1663:
1653:
1652:
1649:
1648:
1633:
1632:
1621:
1608:
1592:Lam, Tsit Yuen
1588:
1575:
1551:
1540:
1537:
1534:
1533:
1521:
1509:
1497:
1467:
1466:
1464:
1461:
1460:
1459:
1448:
1429:
1426:abelian groups
1414:
1403:
1400:
1395:
1391:
1387:
1384:
1379:
1375:
1371:
1366:
1362:
1358:
1338:
1333:
1329:
1325:
1320:
1316:
1312:
1307:
1303:
1299:
1294:
1290:
1286:
1266:
1263:
1258:
1254:
1250:
1247:
1242:
1238:
1234:
1229:
1225:
1221:
1218:
1215:
1210:
1206:
1202:
1197:
1193:
1189:
1184:
1180:
1176:
1171:
1167:
1163:
1160:
1157:
1154:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1106:
1102:
1099:
1086:
1083:
1075:
1074:
1060:
1041:
1028:injective hull
1014:
1011:
1008:
1005:
984:
979:
973:
969:
966:
963:
945:
925:
922:
919:
916:
912:
909:
906:
903:
884:
881:uniform module
877:
866:
842:
839:
836:
833:
828:
823:
820:
817:
812:
809:
805:
802:
798:
793:
788:
785:
782:
752:
749:
736:
733:
730:
727:
724:
721:
717:
713:
710:
707:
702:
698:
661:
657:
653:
575:
572:
569:
566:
561:
557:
525:
521:
517:
488:
485:
482:
477:
472:
469:
466:
423:
420:
417:
412:
407:
404:
401:
359:
356:
353:
348:
343:
340:
337:
302:
298:
295:
291:
286:
281:
278:
275:
270:
267:
264:
261:
256:
251:
248:
245:
219:
216:
195:
192:
189:
185:
181:
178:
175:
170:
166:
146:primary ideals
95:
92:
89:
86:
83:
78:
74:
15:
13:
10:
9:
6:
4:
3:
2:
1683:
1672:
1671:Module theory
1669:
1667:
1664:
1662:
1659:
1658:
1656:
1646:
1640:
1635:
1630:
1626:
1622:
1619:
1615:
1611:
1605:
1601:
1597:
1593:
1589:
1586:
1582:
1578:
1572:
1568:
1564:
1560:
1556:
1552:
1550:
1546:
1543:
1542:
1538:
1531:, p. 86.
1530:
1525:
1522:
1519:, p. 85.
1518:
1513:
1510:
1506:
1501:
1498:
1493:
1489:
1485:
1481:
1480:
1472:
1469:
1462:
1457:
1453:
1449:
1446:
1442:
1438:
1434:
1430:
1427:
1423:
1419:
1415:
1401:
1393:
1389:
1385:
1382:
1377:
1373:
1369:
1364:
1360:
1331:
1327:
1323:
1318:
1314:
1310:
1305:
1301:
1297:
1292:
1288:
1256:
1252:
1248:
1245:
1240:
1236:
1232:
1227:
1223:
1216:
1208:
1204:
1200:
1195:
1191:
1187:
1182:
1178:
1174:
1169:
1165:
1155:
1152:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1100:
1097:
1089:
1088:
1084:
1082:
1080:
1072:
1068:
1065:
1061:
1058:
1029:
1009:
1003:
971:
967:
961:
953:
949:
946:
943:
942:Artinian ring
939:
923:
917:
893:
889:
885:
882:
878:
875:
871:
867:
864:
860:
856:
840:
834:
826:
810:
803:
800:
791:
770:
766:
762:
761:
760:
758:
750:
748:
731:
725:
719:
715:
711:
705:
700:
696:
687:
686:primary ideal
683:
679:
675:
659:
655:
651:
643:
639:
635:
631:
627:
623:
619:
616:
612:
609: ∈
608:
604:
600:
595:
593:
589:
570:
564:
559:
555:
546:
541:
539:
523:
519:
515:
507:
503:
483:
475:
454:
450:
446:
442:
438:
418:
410:
389:
385:
381:
376:
374:
354:
346:
325:
321:
317:
296:
293:
284:
268:
262:
254:
233:
229:
225:
217:
215:
213:
209:
193:
187:
183:
179:
173:
168:
164:
155:
151:
147:
143:
139:
136:
133:of ideals in
132:
128:
123:
121:
113:
109:
93:
87:
81:
76:
72:
63:
59:
55:
51:
48:is a type of
47:
44:
40:
37:
33:
29:
22:
1666:Prime ideals
1628:
1595:
1558:
1548:
1524:
1512:
1500:
1483:
1477:
1471:
1455:
1451:
1444:
1440:
1436:
1432:
1417:
1076:
1070:
1066:
1056:
1026:denotes the
951:
947:
937:
862:
854:
768:
764:
754:
681:
677:
676:. An ideal
673:
644:-primary if
641:
637:
633:
625:
621:
617:
614:
610:
606:
602:
598:
596:
591:
587:
544:
542:
537:
505:
501:
452:
448:
444:
440:
436:
387:
383:
379:
377:
372:
323:
319:
315:
232:prime module
231:
230:is called a
227:
223:
221:
211:
207:
154:prime ideals
141:
124:
119:
112:assassinator
111:
107:
61:
53:
45:
38:
31:
25:
218:Definitions
135:commutative
120:annihilator
58:annihilator
50:prime ideal
1655:Categories
1539:References
751:Properties
640:is called
222:A nonzero
1383:−
1370:−
1246:−
1233:−
1217:⋅
1010:−
890:onto the
811:⊆
706:
599:coprimary
565:
174:
82:
1627:(1970),
1594:(1999),
1557:(1995),
1529:Lam 1999
1517:Lam 1999
1505:Lam 1999
1085:Examples
892:spectrum
804:′
757:Lam 1999
613:implies
386:-module
297:′
226:-module
150:radicals
108:assassin
1618:1653294
1585:1322960
771:, then
41:over a
1616:
1606:
1583:
1573:
1062:For a
996:where
940:is an
857:is an
435:where
382:of an
148:, the
36:module
1463:Notes
680:is a
34:of a
30:, an
1604:ISBN
1571:ISBN
1349:and
1030:and
879:Any
210:and
43:ring
1488:doi
1431:If
1416:If
1090:If
936:If
861:of
855:M'
765:M'
763:If
697:Ass
636:of
601:if
556:Ass
504:of
451:of
378:An
318:of
316:N'
165:Ass
125:In
122:).
114:of
110:or
73:Ass
52:of
26:In
1657::
1614:MR
1612:,
1602:,
1581:MR
1579:,
1569:,
1561:,
1547:,
1484:13
1482:.
1081:.
603:xm
594:.
540:.
375:.
326:,
214:.
1494:.
1490::
1458:.
1456:Z
1452:Z
1447:.
1445:M
1441:M
1437:M
1433:R
1418:R
1402:.
1399:)
1394:3
1390:x
1386:3
1378:3
1374:w
1365:3
1361:z
1357:(
1337:)
1332:2
1328:w
1324:+
1319:2
1315:z
1311:+
1306:2
1302:y
1298:+
1293:2
1289:x
1285:(
1265:)
1262:)
1257:3
1253:x
1249:3
1241:3
1237:w
1228:3
1224:z
1220:(
1214:)
1209:2
1205:w
1201:+
1196:2
1192:z
1188:+
1183:2
1179:y
1175:+
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