1650:
862:
1286:
323:
756:
215:
115:
595:
1358:
509:
444:
380:
1005:
965:, 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
1423:
945:
1064:
1154:
1035:
681:
545:
1159:
785:
1089:
31:
248:
30:
This article mainly concerns associated primes in general ring theory. For the specific usage in commutative ring theory, see also
17:
1618:
1585:
702:
170:
887:
on ideals (for example, any right or left
Noetherian ring) every nonzero module has at least one associated prime.
78:
561:
1291:
1573:
897:
For a one-sided
Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable
469:
404:
340:
1465:(considered as a free module over itself), but its associated prime ideal (2) is not an associated prime of
968:
1671:
1363:
907:
1044:
601:
while the rest of the associated primes (i.e., those properly containing associated primes) are called
141:
1489:
884:
880:
68:
1104:
894:
has either zero or one associated primes, making uniform modules an example of coprimary modules.
1681:
1676:
1010:
1628:
1595:
46:
879:
It is possible, even for a commutative local ring, that the set of associated primes of a
8:
869:
658:
522:
160:
137:
1655:
1635:
1432:
902:
53:
1649:
1614:
1581:
1074:
217:
Also linked with the concept of "associated primes" of the ideal are the notions of
1555:
1498:
898:
145:
38:
1624:
1610:
1591:
1577:
640:
148:
1565:
1487:
Picavet, Gabriel (1985). "Propriétés et applications de la notion de contenu".
1038:
891:
454:. In commutative algebra the usual definition is different, but equivalent: if
1502:
1665:
1602:
1436:
952:
696:
643:
is coprimary if and only if it has exactly one associated prime. A submodule
156:
129:(word play between the notation and the fact that an associated prime is an
164:
60:
32:
Primary decomposition § Primary decomposition from associated primes
1281:{\displaystyle I=((x^{2}+y^{2}+z^{2}+w^{2})\cdot (z^{3}-w^{3}-3x^{3}))}
857:{\displaystyle \mathrm {Ass} _{R}(M')\subseteq \mathrm {Ass} _{R}(M).}
1090:
Primary decomposition#Primary decomposition from associated primes
1080:
over any ring, there are only finitely many associated primes of
318:{\displaystyle \mathrm {Ann} _{R}(N)=\mathrm {Ann} _{R}(N')\,}
1609:, Graduate Texts in Mathematics No. 189, Berlin, New York:
758:; thus, the notion is a generalization of a primary ideal.
597:(with respect to the set-theoretic inclusion) are called
1088:
For the case for commutative
Noetherian rings, see also
1509:
1450:
a finite abelian group, then the associated primes of
766:
Most of these properties and assertions are given in (
75:. The set of associated primes is usually denoted by
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1162:
1107:
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251:
173:
81:
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27:
Prime ideal that is an annihilator a prime submodule
1461:The group of order 2 is a quotient of the integers
751:{\displaystyle \operatorname {Ass} _{R}(R/I)=\{P\}}
1417:
1352:
1280:
1148:
1058:
1029:
999:
939:
856:
750:
675:
589:
539:
503:
438:
374:
317:
209:
109:
1663:
1533:
1521:
883:is empty. However, in any ring satisfying the
210:{\displaystyle \operatorname {Ass} _{R}(R/J).}
167:, and this set of prime ideals coincides with
1454:are exactly the primes dividing the order of
745:
739:
110:{\displaystyle \operatorname {Ass} _{R}(M),}
590:{\displaystyle \operatorname {Ass} _{R}(M)}
1431:is the ring of integers, then non-trivial
155:is decomposed as a finite intersection of
1634:
1353:{\displaystyle (x^{2}+y^{2}+z^{2}+w^{2})}
1115:
1055:
1026:
996:
631: = 0 for some positive integer
500:
435:
371:
314:
1564:
1486:
635:. A nonzero finitely generated module
504:{\displaystyle \mathrm {Ann} _{R}(m)\,}
439:{\displaystyle \mathrm {Ann} _{R}(N)\,}
375:{\displaystyle \mathrm {Ann} _{R}(N)\,}
14:
1664:
1000:{\displaystyle E(R/{\mathfrak {p}})\,}
140:, associated primes are linked to the
1418:{\displaystyle (z^{3}-w^{3}-3x^{3}).}
961:: For a commutative Noetherian ring
955:, then this map becomes a bijection.
458:is commutative, an associated prime
142:Lasker–Noether primary decomposition
1601:
1576:, vol. 150, Berlin, New York:
1539:
1527:
1515:
1439:of prime power order are coprimary.
1050:
988:
940:{\displaystyle \mathrm {Spec} (R).}
876:, their associated primes coincide.
767:
24:
921:
918:
915:
912:
832:
829:
826:
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794:
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481:
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260:
257:
254:
25:
1693:
1059:{\displaystyle {\mathfrak {p}}\,}
1648:
1066:ranges over the prime ideals of
616: = 0 for some nonzero
547:is isomorphic to a submodule of
1156:the associated prime ideals of
1480:
1409:
1367:
1347:
1295:
1275:
1272:
1230:
1224:
1172:
1169:
1149:{\displaystyle R=\mathbb {C} }
1143:
1119:
1023:
1017:
993:
975:
931:
925:
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13:
1:
1607:Lectures on modules and rings
1574:Graduate Texts in Mathematics
1549:
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466:is a prime ideal of the form
1446:is the ring of integers and
163:of these primary ideals are
151:. Specifically, if an ideal
7:
1095:
10:
1698:
325:for any nonzero submodule
71:of a (prime) submodule of
29:
1503:10.1080/00927878508823275
1490:Communications in Algebra
885:ascending chain condition
881:finitely generated module
117:and sometimes called the
1473:
450:is a prime submodule of
401:is an ideal of the form
770:) starting on page 86.
1518:, p. 117, Ex 40B.
1419:
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1031:
1030:{\displaystyle E(-)\,}
1001:
941:
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558:, minimal elements in
554:In a commutative ring
541:
511:for a nonzero element
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211:
111:
18:Associated prime ideal
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333:. For a prime module
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382:is a prime ideal in
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1672:Commutative algebra
1640:Commutative algebra
1636:Matsumura, Hideyuki
1570:Commutative algebra
1560:Algèbre commutative
1433:free abelian groups
870:essential submodule
676:{\displaystyle M/N}
639:over a commutative
608:A module is called
540:{\displaystyle R/P}
245:if the annihilator
138:commutative algebra
1656:Mathematics portal
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1350:
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683:is coprimary with
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315:
207:
107:
67:that arises as an
1620:978-0-387-98428-5
1587:978-0-387-94268-1
1497:(10): 2231–2265.
1075:Noetherian module
899:injective modules
16:(Redirected from
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1658:
1653:
1652:
1642:
1631:
1598:
1556:Nicolas Bourbaki
1543:
1537:
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1513:
1507:
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1435:and non-trivial
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519:or equivalently
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391:associated prime
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216:
214:
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208:
197:
183:
182:
149:Noetherian rings
128:
116:
114:
113:
108:
91:
90:
43:associated prime
39:abstract algebra
21:
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1661:
1654:
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1621:
1611:Springer-Verlag
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1578:Springer-Verlag
1566:Eisenbud, David
1552:
1547:
1546:
1538:
1534:
1526:
1522:
1514:
1510:
1485:
1481:
1476:
1403:
1399:
1387:
1383:
1374:
1370:
1365:
1362:
1361:
1341:
1337:
1328:
1324:
1315:
1311:
1302:
1298:
1293:
1290:
1289:
1288:are the ideals
1266:
1262:
1250:
1246:
1237:
1233:
1218:
1214:
1205:
1201:
1192:
1188:
1179:
1175:
1161:
1158:
1157:
1114:
1106:
1103:
1102:
1098:
1049:
1048:
1046:
1043:
1042:
1012:
1009:
1008:
987:
986:
981:
970:
967:
966:
959:Matlis' Theorem
911:
909:
906:
905:
864:If in addition
836:
825:
824:
810:
801:
790:
789:
787:
784:
783:
764:
725:
710:
706:
704:
701:
700:
699:if and only if
665:
660:
657:
656:
641:Noetherian ring
603:embedded primes
599:isolated primes
569:
565:
563:
560:
559:
529:
524:
521:
520:
485:
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473:
471:
468:
467:
420:
409:
408:
406:
403:
402:
356:
345:
344:
342:
339:
338:
303:
294:
283:
282:
264:
253:
252:
250:
247:
246:
231:
223:embedded primes
219:isolated primes
193:
178:
174:
172:
169:
168:
126:
86:
82:
80:
77:
76:
35:
28:
23:
22:
15:
12:
11:
5:
1695:
1685:
1684:
1679:
1674:
1660:
1659:
1644:
1643:
1632:
1619:
1603:Lam, Tsit Yuen
1599:
1586:
1562:
1551:
1548:
1545:
1544:
1532:
1520:
1508:
1478:
1477:
1475:
1472:
1471:
1470:
1459:
1440:
1437:abelian groups
1425:
1414:
1411:
1406:
1402:
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1395:
1390:
1386:
1382:
1377:
1373:
1369:
1349:
1344:
1340:
1336:
1331:
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1323:
1318:
1314:
1310:
1305:
1301:
1297:
1277:
1274:
1269:
1265:
1261:
1258:
1253:
1249:
1245:
1240:
1236:
1232:
1229:
1226:
1221:
1217:
1213:
1208:
1204:
1200:
1195:
1191:
1187:
1182:
1178:
1174:
1171:
1168:
1165:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1117:
1113:
1110:
1097:
1094:
1086:
1085:
1071:
1052:
1039:injective hull
1025:
1022:
1019:
1016:
995:
990:
984:
980:
977:
974:
956:
936:
933:
930:
927:
923:
920:
917:
914:
895:
892:uniform module
888:
877:
853:
850:
847:
844:
839:
834:
831:
828:
823:
820:
816:
813:
809:
804:
799:
796:
793:
763:
760:
747:
744:
741:
738:
735:
732:
728:
724:
721:
718:
713:
709:
672:
668:
664:
586:
583:
580:
577:
572:
568:
536:
532:
528:
499:
496:
493:
488:
483:
480:
477:
434:
431:
428:
423:
418:
415:
412:
370:
367:
364:
359:
354:
351:
348:
313:
309:
306:
302:
297:
292:
289:
286:
281:
278:
275:
272:
267:
262:
259:
256:
230:
227:
206:
203:
200:
196:
192:
189:
186:
181:
177:
157:primary ideals
106:
103:
100:
97:
94:
89:
85:
26:
9:
6:
4:
3:
2:
1694:
1683:
1682:Module theory
1680:
1678:
1675:
1673:
1670:
1669:
1667:
1657:
1651:
1646:
1641:
1637:
1633:
1630:
1626:
1622:
1616:
1612:
1608:
1604:
1600:
1597:
1593:
1589:
1583:
1579:
1575:
1571:
1567:
1563:
1561:
1557:
1554:
1553:
1542:, p. 86.
1541:
1536:
1530:, p. 85.
1529:
1524:
1517:
1512:
1504:
1500:
1496:
1492:
1491:
1483:
1479:
1468:
1464:
1460:
1457:
1453:
1449:
1445:
1441:
1438:
1434:
1430:
1426:
1412:
1404:
1400:
1396:
1393:
1388:
1384:
1380:
1375:
1371:
1342:
1338:
1334:
1329:
1325:
1321:
1316:
1312:
1308:
1303:
1299:
1267:
1263:
1259:
1256:
1251:
1247:
1243:
1238:
1234:
1227:
1219:
1215:
1211:
1206:
1202:
1198:
1193:
1189:
1185:
1180:
1176:
1166:
1163:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1111:
1108:
1100:
1099:
1093:
1091:
1083:
1079:
1076:
1072:
1069:
1040:
1020:
1014:
982:
978:
972:
964:
960:
957:
954:
953:Artinian ring
950:
934:
928:
904:
900:
896:
893:
889:
886:
882:
878:
875:
871:
867:
851:
845:
837:
821:
814:
811:
802:
781:
777:
773:
772:
771:
769:
759:
742:
736:
730:
726:
722:
716:
711:
707:
698:
697:primary ideal
694:
690:
686:
670:
666:
662:
654:
650:
646:
642:
638:
634:
630:
627:
623:
620: ∈
619:
615:
611:
606:
604:
600:
581:
575:
570:
566:
557:
552:
550:
534:
530:
526:
518:
514:
494:
486:
465:
461:
457:
453:
449:
429:
421:
400:
396:
392:
387:
385:
365:
357:
336:
332:
328:
307:
304:
295:
279:
273:
265:
244:
240:
236:
226:
224:
220:
204:
198:
194:
190:
184:
179:
175:
166:
162:
158:
154:
150:
147:
144:of ideals in
143:
139:
134:
132:
124:
120:
104:
98:
92:
87:
83:
74:
70:
66:
62:
59:is a type of
58:
55:
51:
48:
44:
40:
33:
19:
1677:Prime ideals
1639:
1606:
1569:
1559:
1535:
1523:
1511:
1494:
1488:
1482:
1466:
1462:
1455:
1451:
1447:
1443:
1428:
1087:
1081:
1077:
1067:
1037:denotes the
962:
958:
948:
873:
865:
779:
775:
765:
692:
688:
687:. An ideal
684:
655:-primary if
652:
648:
644:
636:
632:
628:
625:
621:
617:
613:
609:
607:
602:
598:
555:
553:
548:
516:
512:
463:
459:
455:
451:
447:
398:
394:
390:
388:
383:
334:
330:
326:
243:prime module
242:
241:is called a
238:
234:
232:
222:
218:
165:prime ideals
152:
135:
130:
123:assassinator
122:
118:
72:
64:
56:
49:
42:
36:
229:Definitions
146:commutative
131:annihilator
69:annihilator
61:prime ideal
1666:Categories
1550:References
762:Properties
651:is called
233:A nonzero
1394:−
1381:−
1257:−
1244:−
1228:⋅
1021:−
901:onto the
822:⊆
717:
610:coprimary
576:
185:
93:
1638:(1970),
1605:(1999),
1568:(1995),
1540:Lam 1999
1528:Lam 1999
1516:Lam 1999
1096:Examples
903:spectrum
815:′
768:Lam 1999
624:implies
397:-module
308:′
237:-module
161:radicals
119:assassin
1629:1653294
1596:1322960
782:, then
52:over a
1627:
1617:
1594:
1584:
1073:For a
1007:where
951:is an
868:is an
446:where
393:of an
159:, the
47:module
1474:Notes
691:is a
45:of a
41:, an
1615:ISBN
1582:ISBN
1360:and
1041:and
890:Any
221:and
54:ring
1499:doi
1442:If
1427:If
1101:If
947:If
872:of
866:M'
776:M'
774:If
708:Ass
647:of
612:if
567:Ass
515:of
462:of
389:An
329:of
327:N'
176:Ass
136:In
133:).
125:of
121:or
84:Ass
63:of
37:In
1668::
1625:MR
1623:,
1613:,
1592:MR
1590:,
1580:,
1572:,
1558:,
1495:13
1493:.
1092:.
614:xm
605:.
551:.
386:.
337:,
225:.
1505:.
1501::
1469:.
1467:Z
1463:Z
1458:.
1456:M
1452:M
1448:M
1444:R
1429:R
1413:.
1410:)
1405:3
1401:x
1397:3
1389:3
1385:w
1376:3
1372:z
1368:(
1348:)
1343:2
1339:w
1335:+
1330:2
1326:z
1322:+
1317:2
1313:y
1309:+
1304:2
1300:x
1296:(
1276:)
1273:)
1268:3
1264:x
1260:3
1252:3
1248:w
1239:3
1235:z
1231:(
1225:)
1220:2
1216:w
1212:+
1207:2
1203:z
1199:+
1194:2
1190:y
1186:+
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1177:x
1173:(
1170:(
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1144:]
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1135:z
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1126:,
1123:x
1120:[
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1084:.
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1070:.
1068:R
1051:p
1024:)
1018:(
1015:E
994:)
989:p
983:/
979:R
976:(
973:E
963:R
949:R
935:.
932:)
929:R
926:(
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919:e
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852:.
849:)
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830:s
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819:)
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803:R
798:s
795:s
792:A
780:M
778:⊆
746:}
743:P
740:{
737:=
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727:/
723:R
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695:-
693:P
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671:N
667:/
663:M
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649:M
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626:x
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531:/
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335:N
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288:n
285:A
280:=
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274:N
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255:A
239:N
235:R
205:.
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199:J
195:/
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188:(
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153:J
127:M
105:,
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99:M
96:(
88:R
73:M
65:R
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34:.
20:)
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