707:, and so "right singular ring" is not usually defined the same way as singular modules are. Some authors have used "singular ring" to mean "has a nonzero singular ideal", however this usage is not consistent with the usage of the adjectives for modules.
147:
799:
960:
464:
1185:
1034:
255:
705:
650:
1459:
1298:
587:
844:
531:
398:
346:
302:
1114:
184:
1707:
1644:
1558:
1505:
875:
64:
721:
1764:
903:
656:
ring is defined similarly, using the left singular ideal, and it is entirely possible for a ring to be right-but-not-left nonsingular.
403:
1123:
980:
209:
1589:) used nonsingular modules to characterize the class of rings whose maximal right ring of quotients have a certain structure.
663:
1578:
are equivalent: right nonsingular, von
Neumann regular, right semihereditary, right Rickart, Baer, semiprimitive. (
602:
1404:
1247:
1511:
46:
1330:
546:
804:
474:
Here are several definitions used when studying singular submodules and singular ideals. In the following,
496:
363:
311:
267:
1835:
1060:
156:
1840:
1334:
1051:
195:
34:
1665:
1602:
1516:
885:
847:
54:
50:
1468:
1462:
1040:
150:
1807:
1760:
1714:
1230:
1222:
187:
1797:
1752:
1305:
1241:
203:
17:
1819:
1774:
1736:
853:
1815:
1770:
1748:
1732:
1655:
1338:
1326:
1050:
elements of the ring. Consequently, the singular ideal of a commutative ring contains the
1731:, Pure and Applied Mathematics, No. 33, New York: Marcel Dekker Inc., pp. viii+206,
1788:
Zelmanowitz, J. M. (1983), "The structure of rings with faithful nonsingular modules",
1659:
1647:
1564:
Right nonsingularity has a strong interaction with right self injective rings as well.
1382:
1226:
1829:
25:
1346:
1322:
142:{\displaystyle {\mathcal {Z}}(M)=\{m\in M\mid \mathrm {ann} (m)\subseteq _{e}R\}\,}
1352:
For commutative rings, being nonsingular is equivalent to being a reduced ring.
1211:
21:
1756:
1116:, but this does not necessarily hold for the singular submodule. However, if
1811:
1342:
1047:
970:
794:{\displaystyle {\mathcal {Z}}(M_{R})\cdot \mathrm {soc} (R_{R})=\{0\}\,}
1367:, p. 376)) contains several important equivalences. For any ring
1802:
1574:
is a right self injective ring, then the following conditions on
955:{\displaystyle f({\mathcal {Z}}(M))\subseteq {\mathcal {Z}}(N)\,}
1709:
is a finite direct product of full linear rings if and only if
459:{\displaystyle {\mathcal {Z}}(R_{R})\neq {\mathcal {Z}}(_{R}R)}
1253:
1147:
1129:
1011:
986:
937:
915:
727:
669:
608:
552:
502:
435:
409:
369:
317:
273:
233:
162:
70:
1747:, Graduate Texts in Mathematics No. 189, Berlin, New York:
1180:{\displaystyle {\mathcal {Z}}(M/{\mathcal {Z}}(M))=\{0\}\,}
1029:{\displaystyle {\mathcal {Z}}(N)=N\cap {\mathcal {Z}}(M)\,}
715:
Some general properties of the singular submodule include:
1321:
Right nonsingular rings are a very broad class, including
250:{\displaystyle \operatorname {tors} (M)={\mathcal {Z}}(M)}
1668:
1605:
1519:
1471:
1407:
1250:
1126:
1063:
983:
906:
856:
807:
724:
666:
605:
549:
499:
406:
366:
314:
270:
212:
159:
67:
1057:
A general property of the torsion submodule is that
700:{\displaystyle {\mathcal {Z}}(R_{R})\subsetneq R\,}
1701:
1638:
1552:
1499:
1453:
1292:
1179:
1108:
1028:
954:
869:
838:
793:
699:
644:
581:
525:
458:
392:
340:
296:
249:
178:
141:
1218:is right nonsingular, then the converse is true.
1039:The properties "singular" and "nonsingular" are
1713:has a nonsingular, faithful module with finite
660:In rings with unity it is always the case that
1046:The singular ideals of a ring contain central
645:{\displaystyle {\mathcal {Z}}(R_{R})=\{0\}\,}
8:
1493:
1487:
1454:{\displaystyle S=\mathrm {End} (E(R_{R}))\,}
1293:{\displaystyle {\mathcal {Z}}(R_{R})=J(R)\,}
1173:
1167:
1102:
1096:
787:
781:
638:
632:
575:
569:
135:
87:
61:. In set notation it is usually denoted as
1586:
1729:Ring theory: Nonsingular rings and modules
400:is defined similarly. It is possible for
1801:
1684:
1673:
1667:
1621:
1610:
1604:
1535:
1524:
1518:
1496:
1470:
1450:
1438:
1414:
1406:
1289:
1265:
1252:
1251:
1249:
1176:
1146:
1145:
1140:
1128:
1127:
1125:
1105:
1073:
1062:
1025:
1010:
1009:
985:
984:
982:
951:
936:
935:
914:
913:
905:
861:
855:
835:
826:
808:
806:
790:
769:
751:
739:
726:
725:
723:
696:
681:
668:
667:
665:
641:
620:
607:
606:
604:
582:{\displaystyle {\mathcal {Z}}(M)=\{0\}\,}
578:
551:
550:
548:
522:
501:
500:
498:
444:
434:
433:
421:
408:
407:
405:
378:
368:
367:
365:
329:
316:
315:
313:
285:
272:
271:
269:
232:
231:
211:
161:
160:
158:
138:
126:
102:
69:
68:
66:
839:{\displaystyle \mathrm {soc} (M_{R})\,}
526:{\displaystyle {\mathcal {Z}}(M)=M\,}
393:{\displaystyle {\mathcal {Z}}(_{R}R)}
341:{\displaystyle {\mathcal {Z}}(R_{R})}
297:{\displaystyle {\mathcal {Z}}(R_{R})}
7:
308:as a right module, and in this case
1579:
1364:
1421:
1418:
1415:
1120:is a right nonsingular ring, then
815:
812:
809:
758:
755:
752:
194:) which is most often defined for
109:
106:
103:
14:
1109:{\displaystyle t(M/t(M))=\{0\}\,}
179:{\displaystyle {\mathcal {Z}}(M)}
1371:, the following are equivalent:
186:is a good generalization of the
1512:maximal right ring of quotients
1702:{\displaystyle Q_{max}^{r}(R)}
1696:
1690:
1639:{\displaystyle Q_{max}^{r}(R)}
1633:
1627:
1553:{\displaystyle Q_{max}^{r}(R)}
1547:
1541:
1481:
1475:
1447:
1444:
1431:
1425:
1286:
1280:
1271:
1258:
1161:
1158:
1152:
1134:
1090:
1087:
1081:
1067:
1022:
1016:
997:
991:
948:
942:
929:
926:
920:
910:
832:
819:
775:
762:
745:
732:
687:
674:
626:
613:
563:
557:
513:
507:
453:
441:
427:
414:
387:
375:
335:
322:
291:
278:
244:
238:
225:
219:
173:
167:
119:
113:
81:
75:
1:
1745:Lectures on modules and rings
1194:is an essential submodule of
45:consisting of elements whose
1500:{\displaystyle J(S)=\{0\}\,}
1041:Morita invariant properties
360:. The left handed analogue
1857:
1198:(both right modules) then
28:, each right (resp. left)
1757:10.1007/978-1-4612-0525-8
1394:) is a nonsingular right
1331:von Neumann regular rings
1727:Goodearl, K. R. (1976),
348:is a two-sided ideal of
1790:Trans. Amer. Math. Soc.
1743:Lam, Tsit-Yuen (1999),
1560:is von Neumann regular.
1363:(due to R. E. Johnson (
304:is defined considering
1703:
1640:
1554:
1501:
1455:
1401:The endomorphism ring
1327:(semi)hereditary rings
1294:
1181:
1110:
1030:
956:
871:
840:
795:
701:
646:
583:
527:
460:
394:
342:
298:
251:
180:
143:
1704:
1641:
1555:
1502:
1456:
1378:is right nonsingular.
1295:
1182:
1111:
1031:
957:
872:
870:{\displaystyle R_{R}}
841:
796:
702:
647:
584:
528:
461:
395:
343:
299:
252:
181:
144:
1666:
1603:
1517:
1469:
1405:
1248:
1124:
1061:
981:
904:
854:
805:
722:
664:
603:
547:
497:
404:
364:
354:right singular ideal
312:
268:
210:
198:. In the case that
157:
65:
1689:
1654:has a nonsingular,
1626:
1540:
1242:self-injective ring
53:right (resp. left)
16:In the branches of
1699:
1669:
1636:
1606:
1550:
1520:
1497:
1463:semiprimitive ring
1451:
1356:Important theorems
1290:
1177:
1106:
1026:
952:
867:
836:
791:
697:
642:
579:
541:nonsingular module
523:
456:
390:
338:
294:
247:
176:
139:
43:singular submodule
1766:978-0-387-98428-5
1715:uniform dimension
1361:Johnson's Theorem
1231:projective module
1223:semisimple module
1206:is singular. If
597:right nonsingular
188:torsion submodule
1848:
1822:
1805:
1777:
1739:
1708:
1706:
1705:
1700:
1688:
1683:
1648:full linear ring
1645:
1643:
1642:
1637:
1625:
1620:
1599:is a ring, then
1587:Zelmanowitz 1983
1559:
1557:
1556:
1551:
1539:
1534:
1506:
1504:
1503:
1498:
1460:
1458:
1457:
1452:
1443:
1442:
1424:
1339:semisimple rings
1306:Jacobson radical
1299:
1297:
1296:
1291:
1270:
1269:
1257:
1256:
1186:
1184:
1183:
1178:
1151:
1150:
1144:
1133:
1132:
1115:
1113:
1112:
1107:
1077:
1035:
1033:
1032:
1027:
1015:
1014:
990:
989:
961:
959:
958:
953:
941:
940:
919:
918:
876:
874:
873:
868:
866:
865:
845:
843:
842:
837:
831:
830:
818:
800:
798:
797:
792:
774:
773:
761:
744:
743:
731:
730:
706:
704:
703:
698:
686:
685:
673:
672:
654:left nonsingular
651:
649:
648:
643:
625:
624:
612:
611:
588:
586:
585:
580:
556:
555:
532:
530:
529:
524:
506:
505:
465:
463:
462:
457:
449:
448:
439:
438:
426:
425:
413:
412:
399:
397:
396:
391:
383:
382:
373:
372:
347:
345:
344:
339:
334:
333:
321:
320:
303:
301:
300:
295:
290:
289:
277:
276:
256:
254:
253:
248:
237:
236:
185:
183:
182:
177:
166:
165:
148:
146:
145:
140:
131:
130:
112:
74:
73:
37:
18:abstract algebra
1856:
1855:
1851:
1850:
1849:
1847:
1846:
1845:
1826:
1825:
1803:10.2307/1999320
1787:
1784:
1782:Primary sources
1767:
1749:Springer-Verlag
1742:
1726:
1723:
1664:
1663:
1650:if and only if
1601:
1600:
1582:, p. 262)
1515:
1514:
1467:
1466:
1434:
1403:
1402:
1393:
1358:
1319:
1261:
1246:
1245:
1225:is nonsingular
1122:
1121:
1059:
1058:
979:
978:
902:
901:
857:
852:
851:
822:
803:
802:
765:
735:
720:
719:
713:
677:
662:
661:
616:
601:
600:
545:
544:
495:
494:
491:singular module
472:
440:
417:
402:
401:
374:
362:
361:
325:
310:
309:
281:
266:
265:
208:
207:
155:
154:
149:. For general
122:
63:
62:
29:
12:
11:
5:
1854:
1852:
1844:
1843:
1838:
1828:
1827:
1824:
1823:
1796:(1): 347–359,
1783:
1780:
1779:
1778:
1765:
1740:
1722:
1719:
1698:
1695:
1692:
1687:
1682:
1679:
1676:
1672:
1660:uniform module
1635:
1632:
1629:
1624:
1619:
1616:
1613:
1609:
1562:
1561:
1549:
1546:
1543:
1538:
1533:
1530:
1527:
1523:
1508:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1449:
1446:
1441:
1437:
1433:
1430:
1427:
1423:
1420:
1417:
1413:
1410:
1399:
1389:
1383:injective hull
1379:
1357:
1354:
1318:
1315:
1314:
1313:
1288:
1285:
1282:
1279:
1276:
1273:
1268:
1264:
1260:
1255:
1234:
1227:if and only if
1219:
1188:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1149:
1143:
1139:
1136:
1131:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1076:
1072:
1069:
1066:
1055:
1044:
1037:
1024:
1021:
1018:
1013:
1008:
1005:
1002:
999:
996:
993:
988:
963:
950:
947:
944:
939:
934:
931:
928:
925:
922:
917:
912:
909:
892:-modules from
878:
864:
860:
834:
829:
825:
821:
817:
814:
811:
789:
786:
783:
780:
777:
772:
768:
764:
760:
757:
754:
750:
747:
742:
738:
734:
729:
712:
709:
695:
692:
689:
684:
680:
676:
671:
658:
657:
640:
637:
634:
631:
628:
623:
619:
615:
610:
590:
577:
574:
571:
568:
565:
562:
559:
554:
534:
521:
518:
515:
512:
509:
504:
471:
468:
455:
452:
447:
443:
437:
432:
429:
424:
420:
416:
411:
389:
386:
381:
377:
371:
337:
332:
328:
324:
319:
293:
288:
284:
280:
275:
246:
243:
240:
235:
230:
227:
224:
221:
218:
215:
175:
172:
169:
164:
137:
134:
129:
125:
121:
118:
115:
111:
108:
105:
101:
98:
95:
92:
89:
86:
83:
80:
77:
72:
13:
10:
9:
6:
4:
3:
2:
1853:
1842:
1839:
1837:
1836:Module theory
1834:
1833:
1831:
1821:
1817:
1813:
1809:
1804:
1799:
1795:
1791:
1786:
1785:
1781:
1776:
1772:
1768:
1762:
1758:
1754:
1750:
1746:
1741:
1738:
1734:
1730:
1725:
1724:
1720:
1718:
1716:
1712:
1693:
1685:
1680:
1677:
1674:
1670:
1662:. Moreover,
1661:
1657:
1653:
1649:
1630:
1622:
1617:
1614:
1611:
1607:
1598:
1594:
1590:
1588:
1583:
1581:
1577:
1573:
1569:
1565:
1544:
1536:
1531:
1528:
1525:
1521:
1513:
1509:
1490:
1484:
1478:
1472:
1464:
1439:
1435:
1428:
1411:
1408:
1400:
1397:
1392:
1388:
1384:
1380:
1377:
1374:
1373:
1372:
1370:
1366:
1362:
1355:
1353:
1350:
1348:
1347:Rickart rings
1344:
1340:
1336:
1332:
1328:
1324:
1323:reduced rings
1316:
1311:
1307:
1303:
1283:
1277:
1274:
1266:
1262:
1243:
1239:
1235:
1232:
1228:
1224:
1220:
1217:
1213:
1209:
1205:
1201:
1197:
1193:
1189:
1170:
1164:
1155:
1141:
1137:
1119:
1099:
1093:
1084:
1078:
1074:
1070:
1064:
1056:
1053:
1049:
1045:
1042:
1038:
1019:
1006:
1003:
1000:
994:
976:
972:
968:
964:
945:
932:
923:
907:
899:
895:
891:
887:
883:
879:
862:
858:
849:
827:
823:
784:
778:
770:
766:
748:
740:
736:
718:
717:
716:
710:
708:
693:
690:
682:
678:
655:
635:
629:
621:
617:
598:
594:
591:
572:
566:
560:
542:
538:
535:
519:
516:
510:
492:
488:
485:
484:
483:
481:
477:
469:
467:
450:
445:
430:
422:
418:
384:
379:
359:
355:
351:
330:
326:
307:
286:
282:
264:is any ring,
263:
258:
241:
228:
222:
216:
213:
205:
201:
197:
193:
189:
170:
152:
132:
127:
123:
116:
99:
96:
93:
90:
84:
78:
60:
56:
52:
48:
44:
40:
36:
32:
27:
26:module theory
23:
19:
1793:
1789:
1744:
1728:
1710:
1651:
1596:
1592:
1591:
1584:
1575:
1571:
1567:
1566:
1563:
1395:
1390:
1386:
1375:
1368:
1360:
1359:
1351:
1320:
1309:
1301:
1237:
1215:
1207:
1203:
1199:
1195:
1191:
1117:
1054:of the ring.
974:
966:
897:
893:
889:
886:homomorphism
881:
846:denotes the
714:
659:
653:
596:
592:
540:
539:is called a
536:
490:
489:is called a
486:
479:
475:
473:
357:
353:
349:
305:
261:
259:
199:
191:
58:
47:annihilators
42:
38:
30:
15:
1841:Ring theory
1646:is a right
1585:The paper (
1240:is a right
1212:free module
470:Definitions
352:called the
204:commutative
22:ring theory
1830:Categories
1465:(that is,
1345:and right
1343:Baer rings
1300:, where J(
1052:nilradical
711:Properties
595:is called
1812:0002-9947
1721:Textbooks
1304:) is the
1048:nilpotent
1007:∩
971:submodule
933:⊆
749:⋅
691:⊊
482:-module:
431:≠
217:
124:⊆
100:∣
94:∈
51:essential
20:known as
1656:faithful
1593:Theorem:
1580:Lam 1999
1568:Theorem:
1398:-module.
1365:Lam 1999
1325:, right
1317:Examples
1229:it is a
1214:, or if
206:domain,
1820:0697079
1775:1653294
1737:0429962
1335:domains
1244:, then
977:, then
900:, then
196:domains
1818:
1810:
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