848:
677:
981:
752:
228:
Artinian is ambiguous, and it becomes necessary to clarify which module structure is
Artinian. To separate the properties of the two structures, one can abuse terminology and refer to
1190:
1138:
1100:
907:
747:
715:
561:
512:
463:
588:
359:
is automatically an
Artinian bimodule. It may happen, however, that a bimodule is Artinian without its left or right structures being Artinian, as the following example will show.
1212:
583:
1014:
1041:
912:
1214:
module. So, this also provides an example of a faithful
Artinian module over a non-artinian ring. This does not happen for Noetherian case; If
55:, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). Both concepts are named for
749:) is not Noetherian. Yet every descending chain of (without loss of generality) proper submodules terminates: Each such chain has the form
1249:
1414:
1334:
843:{\displaystyle \langle 1/n_{1}\rangle \supseteq \langle 1/n_{2}\rangle \supseteq \langle 1/n_{3}\rangle \supseteq \cdots }
189:
can be considered as a right module, where the action is the natural one given by the ring multiplication on the right.
1302:
672:{\displaystyle \langle 1/p\rangle \subset \langle 1/p^{2}\rangle \subset \langle 1/p^{3}\rangle \subset \cdots }
1237:
Artinian module is also
Noetherian, but over noncommutative rings cyclic Artinian modules can have uncountable
1146:
1105:
1046:
853:
720:
682:
528:
468:
436:
205:
this distinction is necessary, because it is possible for a ring to be
Artinian on one side but not the other.
149:, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring
161:. It also follows that any finitely generated Artinian module is Noetherian even without the assumption of
36:
1270:
142:
63:
1376:
Hartley, B. (1977). "Uncountable
Artinian modules and uncountable soluble groups satisfying Min-n".
331:
whose poset of sub-bimodules satisfies the descending chain condition. Since a sub-bimodule of an
1195:
566:
1433:
377:
be a simple ring which is not right
Artinian. Then it is also not left Artinian. Considering
1252:, which states that the Artinian and Noetherian conditions are equivalent for modules over a
986:
1397:
1368:
1019:
32:
8:
390:
202:
201:
is an
Artinian module. The definition of "left Artinian ring" is done analogously. For
1275:
1238:
158:
52:
279:
243:
The occurrence of modules with a left and right structure is not unusual: for example
232:
as left
Artinian or right Artinian when, strictly speaking, it is correct to say that
1410:
1330:
1265:
523:
426:
208:
The left-right adjectives are not normally necessary for modules, because the module
78:
71:
1102:
is a decreasing sequence of positive integers. Thus the sequence terminates, making
1385:
1356:
1322:
1253:
1230:
106:
24:
1393:
1364:
1280:
370:
294:
146:
94:
67:
1318:
1389:
1360:
1427:
1234:
256:
194:
48:
369:
is left
Artinian if and only if it is right Artinian, in which case it is a
976:{\displaystyle \langle 1/n_{i+1}\rangle \subseteq \langle 1/n_{i}\rangle }
366:
323:
The Artinian condition can be defined on bimodule structures as well: an
20:
515:
406:
180:
56:
1344:
1242:
157:-module is both Noetherian and Artinian, and is said to be of finite
44:
145:
over an Artinian ring is Artinian. Since an Artinian ring is also a
425:
Unlike the case of rings, there are Artinian modules which are not
328:
252:
16:
Module which satisfies the descending chain condition on submodules
1347:(1997). "Cyclic Artinian Modules Without a Composition Series".
1241:
as shown in the article of Hartley and summarized nicely in the
389:-bimodule in the natural way, its sub-bimodules are exactly the
1325:(1969). "Chapter 6. Chain conditions; Chapter 8. Artin rings".
413:
is Artinian as a bimodule, but not Artinian as a left or right
40:
70:, the descending chain condition becomes equivalent to the
81:, Artinian modules enjoy the following heredity property:
174:
1405:
Lam, T.Y. (2001). "Chapter 1. Wedderburn-Artin theory".
420:
1317:
1198:
1149:
1108:
1049:
1022:
989:
915:
856:
755:
723:
685:
591:
569:
531:
471:
439:
278:, it is automatically a left module over the ring of
216:-module at the outset. However, it is possible that
74:, and so that may be used in the definition instead.
251:-module structure. In fact this is an example of a
181:
Left and right Artinian rings, modules and bimodules
1206:
1184:
1132:
1094:
1035:
1008:
975:
901:
842:
741:
709:
671:
577:
555:
506:
457:
1425:
970:
949:
943:
916:
831:
810:
804:
783:
777:
756:
660:
639:
633:
612:
606:
592:
93:-module, then so is any submodule and any
1200:
1185:{\displaystyle \mathbb {Z} /\mathbb {Z} }
1178:
1151:
1133:{\displaystyle \mathbb {Z} (p^{\infty })}
1110:
1095:{\displaystyle n_{1},n_{2},n_{3},\ldots }
902:{\displaystyle n_{1},n_{2},n_{3},\ldots }
742:{\displaystyle \mathbb {Q} /\mathbb {Z} }
735:
725:
710:{\displaystyle \mathbb {Z} (p^{\infty })}
687:
571:
556:{\displaystyle \mathbb {Z} (p^{\infty })}
533:
507:{\displaystyle \mathbb {Z} /\mathbb {Z} }
500:
473:
458:{\displaystyle \mathbb {Q} /\mathbb {Z} }
451:
441:
1297:
1295:
1375:
1245:article dedicated to Hartley's memory.
316:-module, but it is Artinian as a right
1426:
1407:A First Course in Noncommutative Rings
293:-bimodule. For example, consider the
1292:
1218:is a faithful Noetherian module over
1343:
1271:Ascending/Descending chain condition
421:Relation to the Noetherian condition
308:-bimodule in the natural way. Then
224:-module structure, and then calling
212:is usually given as a left or right
1404:
1327:Introduction to Commutative Algebra
13:
1122:
699:
545:
14:
1445:
125:any Artinian submodule such that
62:In the presence of the axiom of (
1250:Akizuki–Hopkins–Levitzki theorem
274:. Indeed, for any right module
255:, and it may be possible for an
240:-module structure, is Artinian.
1248:Another relevant result is the
220:may have both a left and right
1169:
1155:
1127:
1114:
704:
691:
550:
537:
491:
477:
401:is simple there are only two:
270:bimodule for a different ring
1:
1311:
1207:{\displaystyle \mathbb {Z} }
578:{\displaystyle \mathbb {Z} }
429:. For example, consider the
355:-module were Artinian, then
247:itself has a left and right
165:being Artinian. However, if
47:. They are for modules what
7:
1259:
173:is not finitely-generated,
10:
1450:
312:is not Artinian as a left
37:descending chain condition
1361:10.1112/S0024610797004912
175:there are counterexamples
153:, any finitely-generated
143:finitely-generated module
1286:
365:It is well known that a
1390:10.1112/plms/s3-35.1.55
1303:Proposition 1.21, p. 19
1226:is Noetherian as well.
1009:{\displaystyle n_{i+1}}
909:, and the inclusion of
679:does not terminate, so
262:to be made into a left-
197:when this right module
1378:Proc. London Math. Soc
1208:
1186:
1134:
1096:
1037:
1010:
977:
903:
844:
743:
711:
673:
579:
557:
508:
459:
433:-primary component of
141:As a consequence, any
1209:
1187:
1135:
1097:
1038:
1036:{\displaystyle n_{i}}
1011:
978:
904:
845:
744:
712:
674:
580:
558:
509:
460:
417:-module over itself.
409:. Thus the bimodule
351:considered as a left
343:is a fortiori a left
1196:
1147:
1106:
1047:
1020:
987:
913:
854:
753:
721:
683:
589:
567:
529:
469:
437:
285:, and moreover is a
203:noncommutative rings
169:is not Artinian and
1409:. Springer Verlag.
1349:J. London Math. Soc
1192:is also a faithful
585:-module. The chain
35:that satisfies the
1329:. Westview Press.
1276:Composition series
1204:
1182:
1130:
1092:
1033:
1006:
973:
899:
850:for some integers
840:
739:
707:
669:
575:
553:
504:
455:
427:Noetherian modules
133:is Artinian, then
79:Noetherian modules
1416:978-0-387-95325-0
1336:978-0-201-40751-8
1266:Noetherian module
524:quasicyclic group
325:Artinian bimodule
72:minimum condition
1441:
1420:
1401:
1372:
1340:
1306:
1299:
1254:semiprimary ring
1231:commutative ring
1213:
1211:
1210:
1205:
1203:
1191:
1189:
1188:
1183:
1181:
1176:
1165:
1154:
1139:
1137:
1136:
1131:
1126:
1125:
1113:
1101:
1099:
1098:
1093:
1085:
1084:
1072:
1071:
1059:
1058:
1042:
1040:
1039:
1034:
1032:
1031:
1015:
1013:
1012:
1007:
1005:
1004:
982:
980:
979:
974:
969:
968:
959:
942:
941:
926:
908:
906:
905:
900:
892:
891:
879:
878:
866:
865:
849:
847:
846:
841:
830:
829:
820:
803:
802:
793:
776:
775:
766:
748:
746:
745:
740:
738:
733:
728:
716:
714:
713:
708:
703:
702:
690:
678:
676:
675:
670:
659:
658:
649:
632:
631:
622:
602:
584:
582:
581:
576:
574:
562:
560:
559:
554:
549:
548:
536:
513:
511:
510:
505:
503:
498:
487:
476:
464:
462:
461:
456:
454:
449:
444:
295:rational numbers
236:, with its left
193:is called right
25:abstract algebra
1449:
1448:
1444:
1443:
1442:
1440:
1439:
1438:
1424:
1423:
1417:
1337:
1323:Macdonald, I.G.
1314:
1309:
1300:
1293:
1289:
1281:Krull dimension
1262:
1199:
1197:
1194:
1193:
1177:
1172:
1161:
1150:
1148:
1145:
1144:
1121:
1117:
1109:
1107:
1104:
1103:
1080:
1076:
1067:
1063:
1054:
1050:
1048:
1045:
1044:
1027:
1023:
1021:
1018:
1017:
994:
990:
988:
985:
984:
964:
960:
955:
931:
927:
922:
914:
911:
910:
887:
883:
874:
870:
861:
857:
855:
852:
851:
825:
821:
816:
798:
794:
789:
771:
767:
762:
754:
751:
750:
734:
729:
724:
722:
719:
718:
717:(and therefore
698:
694:
686:
684:
681:
680:
654:
650:
645:
627:
623:
618:
598:
590:
587:
586:
570:
568:
565:
564:
544:
540:
532:
530:
527:
526:
499:
494:
483:
472:
470:
467:
466:
450:
445:
440:
438:
435:
434:
423:
371:semisimple ring
183:
147:Noetherian ring
89:is an Artinian
29:Artinian module
23:, specifically
17:
12:
11:
5:
1447:
1437:
1436:
1422:
1421:
1415:
1402:
1373:
1355:(2): 231–235.
1341:
1335:
1313:
1310:
1308:
1307:
1290:
1288:
1285:
1284:
1283:
1278:
1273:
1268:
1261:
1258:
1202:
1180:
1175:
1171:
1168:
1164:
1160:
1157:
1153:
1129:
1124:
1120:
1116:
1112:
1091:
1088:
1083:
1079:
1075:
1070:
1066:
1062:
1057:
1053:
1030:
1026:
1003:
1000:
997:
993:
972:
967:
963:
958:
954:
951:
948:
945:
940:
937:
934:
930:
925:
921:
918:
898:
895:
890:
886:
882:
877:
873:
869:
864:
860:
839:
836:
833:
828:
824:
819:
815:
812:
809:
806:
801:
797:
792:
788:
785:
782:
779:
774:
770:
765:
761:
758:
737:
732:
727:
706:
701:
697:
693:
689:
668:
665:
662:
657:
653:
648:
644:
641:
638:
635:
630:
626:
621:
617:
614:
611:
608:
605:
601:
597:
594:
573:
563:, regarded as
552:
547:
543:
539:
535:
502:
497:
493:
490:
486:
482:
479:
475:
453:
448:
443:
422:
419:
182:
179:
139:
138:
103:
102:
49:Artinian rings
15:
9:
6:
4:
3:
2:
1446:
1435:
1434:Module theory
1432:
1431:
1429:
1418:
1412:
1408:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1374:
1370:
1366:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1332:
1328:
1324:
1320:
1316:
1315:
1304:
1298:
1296:
1291:
1282:
1279:
1277:
1274:
1272:
1269:
1267:
1264:
1263:
1257:
1255:
1251:
1246:
1244:
1240:
1236:
1232:
1227:
1225:
1221:
1217:
1173:
1166:
1162:
1158:
1141:
1118:
1089:
1086:
1081:
1077:
1073:
1068:
1064:
1060:
1055:
1051:
1028:
1024:
1001:
998:
995:
991:
983:implies that
965:
961:
956:
952:
946:
938:
935:
932:
928:
923:
919:
896:
893:
888:
884:
880:
875:
871:
867:
862:
858:
837:
834:
826:
822:
817:
813:
807:
799:
795:
790:
786:
780:
772:
768:
763:
759:
730:
695:
666:
663:
655:
651:
646:
642:
636:
628:
624:
619:
615:
609:
603:
599:
595:
541:
525:
521:
517:
495:
488:
484:
480:
446:
432:
428:
418:
416:
412:
408:
404:
400:
396:
392:
388:
384:
380:
376:
372:
368:
364:
360:
358:
354:
350:
346:
342:
338:
334:
330:
326:
321:
319:
315:
311:
307:
303:
299:
296:
292:
288:
284:
281:
277:
273:
269:
265:
261:
258:
257:abelian group
254:
250:
246:
241:
239:
235:
231:
227:
223:
219:
215:
211:
206:
204:
200:
196:
192:
188:
178:
176:
172:
168:
164:
160:
156:
152:
148:
144:
136:
132:
128:
124:
120:
116:
112:
111:
110:
108:
100:
96:
92:
88:
84:
83:
82:
80:
75:
73:
69:
65:
60:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
1406:
1384:(1): 55–75.
1381:
1380:. Series 3.
1377:
1352:
1351:. Series 2.
1348:
1326:
1319:Atiyah, M.F.
1301:Lam (2001),
1247:
1228:
1223:
1219:
1215:
1142:
1016:must divide
519:
430:
424:
414:
410:
402:
398:
394:
386:
382:
378:
374:
362:
361:
356:
352:
348:
347:-module, if
344:
340:
336:
332:
324:
322:
317:
313:
309:
305:
301:
297:
290:
286:
282:
275:
271:
267:
263:
259:
248:
244:
242:
237:
233:
229:
225:
221:
217:
213:
209:
207:
198:
190:
186:
184:
170:
166:
162:
154:
150:
140:
137:is Artinian.
134:
130:
126:
122:
121:-module and
118:
114:
109:also holds:
104:
98:
90:
86:
76:
61:
28:
18:
514:, which is
367:simple ring
21:mathematics
1345:Cohn, P.M.
1312:References
1143:Note that
1140:Artinian.
516:isomorphic
465:, that is
407:zero ideal
339:-bimodule
57:Emil Artin
45:submodules
1243:Paul Cohn
1123:∞
1090:…
971:⟩
950:⟨
947:⊆
944:⟩
917:⟨
897:…
838:⋯
835:⊇
832:⟩
811:⟨
808:⊇
805:⟩
784:⟨
781:⊇
778:⟩
757:⟨
700:∞
667:⋯
664:⊂
661:⟩
640:⟨
637:⊂
634:⟩
613:⟨
610:⊂
607:⟩
593:⟨
546:∞
320:-module.
185:The ring
64:dependent
1428:Category
1260:See also
1233:, every
405:and the
397:. Since
363:Example:
329:bimodule
280:integers
266:, right-
253:bimodule
195:Artinian
107:converse
95:quotient
51:are for
1398:0442091
1369:1438626
1229:Over a
518:to the
373:. Let
117:is any
39:on its
1413:
1396:
1367:
1333:
1239:length
1235:cyclic
391:ideals
381:as an
159:length
68:choice
33:module
1287:Notes
1222:then
1043:. So
327:is a
300:as a
77:Like
53:rings
41:poset
31:is a
27:, an
1411:ISBN
1331:ISBN
105:The
1386:doi
1357:doi
393:of
113:If
97:of
85:If
43:of
19:In
1430::
1394:MR
1392:.
1382:35
1365:MR
1363:.
1353:55
1321:;
1294:^
1256:.
177:.
66:)
59:.
1419:.
1400:.
1388::
1371:.
1359::
1339:.
1305:.
1224:A
1220:A
1216:M
1201:Z
1179:Z
1174:/
1170:]
1167:p
1163:/
1159:1
1156:[
1152:Z
1128:)
1119:p
1115:(
1111:Z
1087:,
1082:3
1078:n
1074:,
1069:2
1065:n
1061:,
1056:1
1052:n
1029:i
1025:n
1002:1
999:+
996:i
992:n
966:i
962:n
957:/
953:1
939:1
936:+
933:i
929:n
924:/
920:1
894:,
889:3
885:n
881:,
876:2
872:n
868:,
863:1
859:n
827:3
823:n
818:/
814:1
800:2
796:n
791:/
787:1
773:1
769:n
764:/
760:1
736:Z
731:/
726:Q
705:)
696:p
692:(
688:Z
656:3
652:p
647:/
643:1
629:2
625:p
620:/
616:1
604:p
600:/
596:1
572:Z
551:)
542:p
538:(
534:Z
522:-
520:p
501:Z
496:/
492:]
489:p
485:/
481:1
478:[
474:Z
452:Z
447:/
442:Q
431:p
415:R
411:R
403:R
399:R
395:R
387:R
385:-
383:R
379:R
375:R
357:M
353:R
349:M
345:R
341:M
337:S
335:-
333:R
318:Q
314:Z
310:Q
306:Q
304:-
302:Z
298:Q
291:R
289:-
287:Z
283:Z
276:M
272:S
268:S
264:R
260:M
249:R
245:R
238:R
234:M
230:M
226:M
222:R
218:M
214:R
210:M
199:R
191:R
187:R
171:M
167:R
163:R
155:R
151:R
135:M
131:N
129:/
127:M
123:N
119:R
115:M
101:.
99:M
91:R
87:M
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