1402:
Proof: first, a strictly increasing sequence cannot stabilize, obviously. Conversely, suppose there is an ascending sequence that does not stabilize; then clearly it contains a strictly increasing (necessarily infinite)
562:
699:
216:
385:
814:
144:
304:
1304:
1083:
1348:
1326:
1105:
1021:
959:
587:
1213:
1186:
1159:
1132:
1233:
999:
979:
937:
917:
897:
877:
857:
837:
722:
607:
1368:
461:
Every finite poset satisfies both the ascending and descending chain conditions, and thus is both well-founded and converse well-founded.
476:
615:
1577:
1595:"Is the equivalence of the ascending chain condition and the maximum condition equivalent to the axiom of dependent choice?"
159:
48:. These conditions played an important role in the development of the structure theory of commutative rings in the works of
331:
730:
1485:
90:
1556:
1537:
1519:
1494:
235:
1378:
1238:
1029:
1611:
1511:
1363:
404:
318:
64:. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.
1621:
1331:
1309:
1088:
1004:
942:
570:
60:. The conditions themselves can be stated in an abstract form, so that they make sense for any
1616:
1306:. That is, after some point all the ideals are equal to each other. Therefore, the ideals of
73:
61:
1191:
1164:
1137:
1110:
8:
446:
being converse well-founded (again, assuming dependent choice): every nonempty subset of
41:
37:
1417:
1328:
satisfy the ascending chain condition, where ideals are ordered by set inclusion. Hence
1503:
1218:
984:
964:
922:
902:
882:
862:
842:
822:
707:
592:
1573:
1552:
1533:
1515:
1490:
45:
1565:
1373:
1351:
1605:
49:
1594:
1480:
1001:. However, at this point there is no larger ideal; we have "topped out" at
416:
53:
432:
17:
1429:
1383:
436:
57:
225:
eventually stabilizes, meaning that there exists a positive integer
557:{\displaystyle \mathbb {Z} =\{\dots ,-3,-2,-1,0,1,2,3,\dots \}}
407:, the descending chain condition on (possibly infinite) poset
1502:
1423:
1441:
442:
Similarly, the ascending chain condition is equivalent to
694:{\displaystyle I=\{\dots ,-18,-12,-6,0,6,12,18,\dots \}}
1458:
1456:
153:
exists. Equivalently, every weakly ascending sequence
211:{\displaystyle a_{1}\leq a_{2}\leq a_{3}\leq \cdots ,}
1334:
1312:
1241:
1221:
1194:
1167:
1140:
1113:
1091:
1032:
1007:
987:
967:
945:
925:
905:
885:
865:
845:
825:
733:
710:
618:
595:
573:
479:
380:{\displaystyle a_{1}\geq a_{2}\geq a_{3}\geq \cdots }
334:
238:
162:
93:
1453:
809:{\displaystyle J=\{\dots ,-6,-4,-2,0,2,4,6,\dots \}}
1551:(5th ed.), Addison-Wesley Publishing Company,
1342:
1320:
1298:
1227:
1207:
1180:
1153:
1126:
1099:
1077:
1015:
993:
973:
953:
931:
911:
891:
871:
851:
831:
808:
716:
693:
601:
581:
556:
379:
298:
210:
139:{\displaystyle a_{1}<a_{2}<a_{3}<\cdots }
138:
84:(ACC) if no infinite strictly ascending sequence
325:. Equivalently, every weakly descending sequence
1603:
1479:
1506:; Gubareni, Nadiya; Kirichenko, V. V. (2004),
1369:Ascending chain condition for principal ideals
299:{\displaystyle a_{n}=a_{n+1}=a_{n+2}=\cdots .}
36:) are finiteness properties satisfied by some
1299:{\displaystyle I_{n}=I_{n+1}=I_{n+2}=\cdots }
1546:
1435:
819:be the ideal consisting of all multiples of
803:
740:
688:
625:
551:
488:
1547:Fraleigh, John B.; Katz, Victor J. (1967),
1424:Hazewinkel, Gubareni & Kirichenko 2004
1336:
1314:
1093:
1009:
947:
589:consists of all multiples of some number
575:
481:
1564:
1447:
1078:{\displaystyle I_{1},I_{2},I_{3},\dots }
423:has a minimal element (also called the
1604:
1527:
1462:
1486:Introduction to Commutative Algebra
13:
1549:A first course in abstract algebra
14:
1633:
1587:
1379:Maximal condition on congruences
1215:, and so on, then there is some
1396:
859:is contained inside the ideal
1:
1472:
704:consists of all multiples of
67:
1530:Encyclopaedia of Mathematics
1410:
1343:{\displaystyle \mathbb {Z} }
1321:{\displaystyle \mathbb {Z} }
1100:{\displaystyle \mathbb {Z} }
1016:{\displaystyle \mathbb {Z} }
954:{\displaystyle \mathbb {Z} }
582:{\displaystyle \mathbb {Z} }
7:
1508:Algebras, rings and modules
1483:; MacDonald, I. G. (1969),
1357:
567:of integers. Each ideal of
450:has a maximal element (the
419:: every nonempty subset of
397:
10:
1638:
1512:Kluwer Academic Publishers
961:, since every multiple of
939:is contained in the ideal
879:, since every multiple of
465:
435:that is well-founded is a
315:descending chain condition
30:descending chain condition
609:. For example, the ideal
405:axiom of dependent choice
319:infinite descending chain
82:ascending chain condition
22:ascending chain condition
1436:Fraleigh & Katz 1967
1389:
394:eventually stabilizes.
313:is said to satisfy the
80:is said to satisfy the
1344:
1322:
1300:
1229:
1209:
1182:
1155:
1128:
1101:
1079:
1017:
995:
975:
955:
933:
913:
899:is also a multiple of
893:
873:
853:
833:
810:
718:
695:
603:
583:
558:
381:
300:
212:
140:
1528:Hazewinkel, Michiel.
1345:
1323:
1301:
1230:
1210:
1208:{\displaystyle I_{3}}
1183:
1181:{\displaystyle I_{2}}
1156:
1154:{\displaystyle I_{2}}
1129:
1127:{\displaystyle I_{1}}
1102:
1080:
1018:
996:
976:
956:
934:
919:. In turn, the ideal
914:
894:
874:
854:
834:
811:
719:
696:
604:
584:
559:
382:
317:(DCC) if there is no
301:
213:
141:
74:partially ordered set
62:partially ordered set
1332:
1310:
1239:
1219:
1192:
1165:
1138:
1111:
1089:
1030:
1005:
985:
965:
943:
923:
903:
883:
863:
843:
823:
731:
708:
616:
593:
571:
477:
332:
236:
160:
91:
38:algebraic structures
1612:Commutative algebra
1504:Hazewinkel, Michiel
1450:, pp. 142, 147
1438:, p. 366, Lemma 7.1
1426:, p. 6, Prop. 1.1.4
433:totally ordered set
40:, most importantly
1340:
1318:
1296:
1225:
1205:
1178:
1151:
1124:
1097:
1075:
1013:
991:
971:
951:
929:
909:
889:
869:
849:
829:
806:
714:
691:
599:
579:
554:
470:Consider the ring
377:
296:
208:
136:
1579:978-0-486-47189-1
1489:, Perseus Books,
1228:{\displaystyle n}
994:{\displaystyle 1}
981:is a multiple of
974:{\displaystyle 2}
932:{\displaystyle J}
912:{\displaystyle 2}
892:{\displaystyle 6}
872:{\displaystyle J}
852:{\displaystyle I}
832:{\displaystyle 2}
717:{\displaystyle 6}
602:{\displaystyle n}
456:maximum condition
452:maximal condition
429:minimum condition
425:minimal condition
411:is equivalent to
46:commutative rings
1629:
1598:
1582:
1566:Jacobson, Nathan
1561:
1543:
1524:
1499:
1466:
1460:
1451:
1445:
1439:
1433:
1427:
1421:
1404:
1400:
1349:
1347:
1346:
1341:
1339:
1327:
1325:
1324:
1319:
1317:
1305:
1303:
1302:
1297:
1289:
1288:
1270:
1269:
1251:
1250:
1234:
1232:
1231:
1226:
1214:
1212:
1211:
1206:
1204:
1203:
1188:is contained in
1187:
1185:
1184:
1179:
1177:
1176:
1160:
1158:
1157:
1152:
1150:
1149:
1134:is contained in
1133:
1131:
1130:
1125:
1123:
1122:
1106:
1104:
1103:
1098:
1096:
1084:
1082:
1081:
1076:
1068:
1067:
1055:
1054:
1042:
1041:
1022:
1020:
1019:
1014:
1012:
1000:
998:
997:
992:
980:
978:
977:
972:
960:
958:
957:
952:
950:
938:
936:
935:
930:
918:
916:
915:
910:
898:
896:
895:
890:
878:
876:
875:
870:
858:
856:
855:
850:
838:
836:
835:
830:
815:
813:
812:
807:
723:
721:
720:
715:
700:
698:
697:
692:
608:
606:
605:
600:
588:
586:
585:
580:
578:
563:
561:
560:
555:
484:
437:well-ordered set
386:
384:
383:
378:
370:
369:
357:
356:
344:
343:
305:
303:
302:
297:
286:
285:
267:
266:
248:
247:
217:
215:
214:
209:
198:
197:
185:
184:
172:
171:
145:
143:
142:
137:
129:
128:
116:
115:
103:
102:
1637:
1636:
1632:
1631:
1630:
1628:
1627:
1626:
1622:Wellfoundedness
1602:
1601:
1593:
1590:
1585:
1580:
1570:Basic Algebra I
1559:
1540:
1522:
1497:
1475:
1470:
1469:
1461:
1454:
1446:
1442:
1434:
1430:
1422:
1418:
1413:
1408:
1407:
1401:
1397:
1392:
1374:Krull dimension
1360:
1352:Noetherian ring
1335:
1333:
1330:
1329:
1313:
1311:
1308:
1307:
1278:
1274:
1259:
1255:
1246:
1242:
1240:
1237:
1236:
1220:
1217:
1216:
1199:
1195:
1193:
1190:
1189:
1172:
1168:
1166:
1163:
1162:
1145:
1141:
1139:
1136:
1135:
1118:
1114:
1112:
1109:
1108:
1092:
1090:
1087:
1086:
1063:
1059:
1050:
1046:
1037:
1033:
1031:
1028:
1027:
1026:In general, if
1008:
1006:
1003:
1002:
986:
983:
982:
966:
963:
962:
946:
944:
941:
940:
924:
921:
920:
904:
901:
900:
884:
881:
880:
864:
861:
860:
844:
841:
840:
824:
821:
820:
732:
729:
728:
709:
706:
705:
617:
614:
613:
594:
591:
590:
574:
572:
569:
568:
480:
478:
475:
474:
468:
400:
390:of elements of
365:
361:
352:
348:
339:
335:
333:
330:
329:
321:of elements of
275:
271:
256:
252:
243:
239:
237:
234:
233:
221:of elements of
193:
189:
180:
176:
167:
163:
161:
158:
157:
149:of elements of
124:
120:
111:
107:
98:
94:
92:
89:
88:
70:
12:
11:
5:
1635:
1625:
1624:
1619:
1614:
1600:
1599:
1589:
1588:External links
1586:
1584:
1583:
1578:
1562:
1557:
1544:
1538:
1525:
1520:
1500:
1495:
1476:
1474:
1471:
1468:
1467:
1452:
1440:
1428:
1415:
1414:
1412:
1409:
1406:
1405:
1394:
1393:
1391:
1388:
1387:
1386:
1381:
1376:
1371:
1366:
1359:
1356:
1338:
1316:
1295:
1292:
1287:
1284:
1281:
1277:
1273:
1268:
1265:
1262:
1258:
1254:
1249:
1245:
1235:for which all
1224:
1202:
1198:
1175:
1171:
1148:
1144:
1121:
1117:
1095:
1085:are ideals of
1074:
1071:
1066:
1062:
1058:
1053:
1049:
1045:
1040:
1036:
1011:
990:
970:
949:
928:
908:
888:
868:
848:
828:
817:
816:
805:
802:
799:
796:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
713:
702:
701:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
627:
624:
621:
598:
577:
565:
564:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
487:
483:
467:
464:
463:
462:
459:
440:
399:
396:
388:
387:
376:
373:
368:
364:
360:
355:
351:
347:
342:
338:
307:
306:
295:
292:
289:
284:
281:
278:
274:
270:
265:
262:
259:
255:
251:
246:
242:
219:
218:
207:
204:
201:
196:
192:
188:
183:
179:
175:
170:
166:
147:
146:
135:
132:
127:
123:
119:
114:
110:
106:
101:
97:
69:
66:
9:
6:
4:
3:
2:
1634:
1623:
1620:
1618:
1615:
1613:
1610:
1609:
1607:
1596:
1592:
1591:
1581:
1575:
1571:
1567:
1563:
1560:
1558:0-201-53467-3
1554:
1550:
1545:
1541:
1539:1-55608-010-7
1535:
1531:
1526:
1523:
1521:1-4020-2690-0
1517:
1513:
1509:
1505:
1501:
1498:
1496:0-201-00361-9
1492:
1488:
1487:
1482:
1481:Atiyah, M. F.
1478:
1477:
1465:, p. 580
1464:
1459:
1457:
1449:
1448:Jacobson 2009
1444:
1437:
1432:
1425:
1420:
1416:
1399:
1395:
1385:
1382:
1380:
1377:
1375:
1372:
1370:
1367:
1365:
1362:
1361:
1355:
1353:
1293:
1290:
1285:
1282:
1279:
1275:
1271:
1266:
1263:
1260:
1256:
1252:
1247:
1243:
1222:
1200:
1196:
1173:
1169:
1146:
1142:
1119:
1115:
1072:
1069:
1064:
1060:
1056:
1051:
1047:
1043:
1038:
1034:
1024:
988:
968:
926:
906:
886:
866:
846:
826:
800:
797:
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
737:
734:
727:
726:
725:
711:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
622:
619:
612:
611:
610:
596:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
485:
473:
472:
471:
460:
457:
453:
449:
445:
441:
438:
434:
430:
426:
422:
418:
414:
410:
406:
403:Assuming the
402:
401:
395:
393:
374:
371:
366:
362:
358:
353:
349:
345:
340:
336:
328:
327:
326:
324:
320:
316:
312:
293:
290:
287:
282:
279:
276:
272:
268:
263:
260:
257:
253:
249:
244:
240:
232:
231:
230:
228:
224:
205:
202:
199:
194:
190:
186:
181:
177:
173:
168:
164:
156:
155:
154:
152:
133:
130:
125:
121:
117:
112:
108:
104:
99:
95:
87:
86:
85:
83:
79:
75:
65:
63:
59:
55:
51:
50:David Hilbert
47:
43:
39:
35:
31:
27:
23:
19:
1617:Order theory
1569:
1548:
1529:
1507:
1484:
1443:
1431:
1419:
1403:subsequence.
1398:
1025:
839:. The ideal
818:
703:
566:
469:
455:
451:
447:
443:
428:
424:
420:
417:well-founded
412:
408:
391:
389:
322:
314:
310:
308:
226:
222:
220:
150:
148:
81:
77:
71:
54:Emmy Noether
33:
29:
25:
21:
15:
309:Similarly,
44:in certain
18:mathematics
1606:Categories
1532:. Kluwer.
1473:References
1463:Hazewinkel
1384:Noetherian
1107:such that
229:such that
68:Definition
58:Emil Artin
1572:, Dover,
1411:Citations
1294:⋯
1073:…
801:…
768:−
759:−
750:−
744:…
686:…
653:−
644:−
635:−
629:…
549:…
516:−
507:−
498:−
492:…
375:⋯
372:≥
359:≥
346:≥
291:⋯
203:⋯
200:≤
187:≤
174:≤
134:⋯
1568:(2009),
1364:Artinian
1358:See also
398:Comments
76:(poset)
466:Example
1576:
1555:
1536:
1518:
1493:
724:. Let
415:being
56:, and
42:ideals
28:) and
20:, the
1390:Notes
1350:is a
431:). A
1574:ISBN
1553:ISBN
1534:ISBN
1516:ISBN
1491:ISBN
131:<
118:<
105:<
454:or
427:or
34:DCC
26:ACC
16:In
1608::
1514:,
1510:,
1455:^
1354:.
1161:,
1023:.
680:18
674:12
647:12
638:18
458:).
72:A
52:,
1597:.
1542:.
1337:Z
1315:Z
1291:=
1286:2
1283:+
1280:n
1276:I
1272:=
1267:1
1264:+
1261:n
1257:I
1253:=
1248:n
1244:I
1223:n
1201:3
1197:I
1174:2
1170:I
1147:2
1143:I
1120:1
1116:I
1094:Z
1070:,
1065:3
1061:I
1057:,
1052:2
1048:I
1044:,
1039:1
1035:I
1010:Z
989:1
969:2
948:Z
927:J
907:2
887:6
867:J
847:I
827:2
804:}
798:,
795:6
792:,
789:4
786:,
783:2
780:,
777:0
774:,
771:2
765:,
762:4
756:,
753:6
747:,
741:{
738:=
735:J
712:6
689:}
683:,
677:,
671:,
668:6
665:,
662:0
659:,
656:6
650:,
641:,
632:,
626:{
623:=
620:I
597:n
576:Z
552:}
546:,
543:3
540:,
537:2
534:,
531:1
528:,
525:0
522:,
519:1
513:,
510:2
504:,
501:3
495:,
489:{
486:=
482:Z
448:P
444:P
439:.
421:P
413:P
409:P
392:P
367:3
363:a
354:2
350:a
341:1
337:a
323:P
311:P
294:.
288:=
283:2
280:+
277:n
273:a
269:=
264:1
261:+
258:n
254:a
250:=
245:n
241:a
227:n
223:P
206:,
195:3
191:a
182:2
178:a
169:1
165:a
151:P
126:3
122:a
113:2
109:a
100:1
96:a
78:P
32:(
24:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.