1281:
644:
1443:. In some sense, this more closely mimics the properties of commutative von Neumann regular rings, which are subdirect products of fields. For commutative rings, von Neumann regular and strongly von Neumann regular are equivalent. In general, the following are equivalent for a ring
467:
423:
1209:
1150:
639:{\displaystyle AXA=U{\begin{pmatrix}I_{r}&0\\0&0\end{pmatrix}}{\begin{pmatrix}I_{r}&0\\0&0\end{pmatrix}}V=U{\begin{pmatrix}I_{r}&0\\0&0\end{pmatrix}}V=A.}
229:
201:
169:
1435:. The condition is left-right symmetric. Strongly von Neumann regular rings are unit regular. Every strongly von Neumann regular ring is a
356:
1158:
1302:
1695:
1570:
1474:
1100:
1797:
1638:
1328:
1310:
813:
1306:
1705:
1611:
Burklund, Robert; Schlank, Tomer M.; Yuan, Allen (2022-07-20). "The
Chromatic Nullstellensatz". p. 50.
1685:
1700:
1830:
1860:
819:
210:
182:
150:
1291:
1295:
279:
1255:
1066:
998:
1855:
1807:
1724:
1395:
1056:
901:
336:
81:
1838:
1815:
1772:
1746:
1648:
8:
1492:
1153:
763:
759:
250:
119:
107:
103:
1728:
1755:
1661:
1612:
1500:
1259:
1240:
928:
115:
28:
1792:(2 ed.), Malabar, FL: Robert E. Krieger Publishing Co. Inc., pp. xviii+412,
1793:
1760:
1675:
1656:
1634:
1512:
1436:
854:
734:
679:
340:
1834:
1822:
1811:
1768:
1750:
1742:
1732:
1712:
1680:
1670:
1644:
1626:
1496:
1248:
1095:
924:
315:
95:
1633:, Chicago lectures in mathematics (Second ed.), University of Chicago Press,
1803:
1391:
987:
806:
752:
748:
445:
275:
285:
Another important class of examples of von
Neumann regular rings are the rings M
893:
826:
34:(associative, with 1, not necessarily commutative) such that for every element
1849:
1440:
1006:
671:
303:
172:
659:
matrix ring over any von
Neumann regular ring is again von Neumann regular.
1764:
1737:
1517:
1461:
1015:
is a subring of a product of fields closed under taking "weak inverses" of
991:
770:
667:
111:
876:
88:
20:
909:
809:
751:
is von
Neumann regular. Indeed, the semisimple rings are precisely the
729:
254:
110:. Von Neumann regular rings should not be confused with the unrelated
916:
The corresponding statements for right modules are also equivalent to
418:{\displaystyle A=U{\begin{pmatrix}I_{r}&0\\0&0\end{pmatrix}}V}
1350:
850:
712:
1398:. An ordinary von Neumann regular ring need not be directly finite.
1280:
839:
every finitely generated left ideal is a direct summand of the left
80:, because these rings are characterized by the fact that every left
1617:
102:) under the name of "regular rings", in the course of his study of
969:
The following statements are equivalent for the commutative ring
1204:{\displaystyle \{0\}\sqcup \mathbb {G} _{m}\to \mathbb {A} ^{1}}
1588:
962:, so there is a canonical way to choose the "weak inverse" of
1577:
934:
In a commutative von
Neumann regular ring, for each element
278:
is von
Neumann regular if and only if it is a field. Every
1215:
Also, the following are equivalent: for a commutative ring
282:
of von
Neumann regular rings is again von Neumann regular.
1537:
1535:
1533:
1547:
1530:
1270:
1103:
586:
534:
491:
374:
1559:
1487:
Generalizations of von
Neumann regular rings include
1161:
791:
The following statements are equivalent for the ring
470:
359:
213:
185:
153:
1610:
1594:
1341:Special types of von Neumann regular rings include
1145:{\textstyle \mathrm {Spec} (R)\to \mathbb {A} ^{1}}
1203:
1144:
638:
417:
223:
195:
163:
1654:
1583:
1847:
1686:20.500.12110/paper_00218693_v25_n1_p185_Michler
1469:is von Neumann regular and every idempotent in
747:) is von Neumann regular. In particular, every
1655:Michler, G.O.; Villamayor, O.E. (April 1973).
1657:"On rings whose simple modules are injective"
1569:sfn error: no target: CITEREFSkornyakov2001 (
783:. Every Boolean ring is von Neumann regular.
94:Von Neumann regular rings were introduced by
76:. Von Neumann regular rings are also called
1696:"Regular ring (in the sense of von Neumann)"
1394:is unit regular, and unit regular rings are
1168:
1162:
1821:
1711:
1309:. Unsourced material may be challenged and
773:is a ring in which every element satisfies
99:
1693:
1565:
1754:
1736:
1684:
1674:
1625:
1616:
1553:
1541:
1329:Learn how and when to remove this message
1191:
1176:
1132:
699:Generalizing the above examples, suppose
1787:
822:left ideal is generated by an idempotent
1848:
931:(also called "Jacobson semi-simple").
1483:is generated by a central idempotent
1307:adding citations to reliable sources
1274:
1271:Generalizations and specializations
923:Every von Neumann regular ring has
216:
188:
156:
64:as a "weak inverse" of the element
13:
1780:
1347:strongly von Neumann regular rings
1114:
1111:
1108:
1105:
692:) is von Neumann regular, even if
14:
1872:
1595:Burklund, Schlank & Yuan 2022
1279:
825:every principal left ideal is a
1454:is strongly von Neumann regular
1094:, or said geometrically, every
224:{\displaystyle {\mathfrak {i}}}
196:{\displaystyle {\mathfrak {i}}}
164:{\displaystyle {\mathfrak {i}}}
1479:Every principal left ideal of
1186:
1127:
1124:
1118:
1069:against the ring homomorphism
257:) is von Neumann regular: for
72:is not uniquely determined by
1:
1603:
1584:Michler & Villamayor 1973
306:with entries from some field
16:Rings admitting weak inverses
1715:(1936), "On Regular Rings",
1676:10.1016/0021-8693(73)90088-4
1407:strongly von Neumann regular
7:
1701:Encyclopedia of Mathematics
1506:
1491:-regular rings, left/right
1460:is von Neumann regular and
920:being von Neumann regular.
755:von Neumann regular rings.
696:is not finite-dimensional.
244:
131:von Neumann regular element
10:
1877:
1831:Princeton University Press
1717:Proc. Natl. Acad. Sci. USA
1694:Skornyakov, L.A. (2001) ,
938:there is a unique element
171:is called a (von Neumann)
1790:von Neumann regular rings
1258:and Zariski topology for
849:every finitely generated
1788:Goodearl, K. R. (1991),
1523:
879:: this is also known as
786:
766:is von Neumann regular.
711:-module such that every
203:there exists an element
25:von Neumann regular ring
1236:is von Neumann regular.
980:is von Neumann regular.
865:is a direct summand of
719:is a direct summand of
1738:10.1073/pnas.22.12.707
1256:constructible topology
1205:
1146:
1067:right-lifting property
802:is von Neumann regular
640:
419:
225:
197:
165:
129:of a ring is called a
1827:Continuous geometries
1396:directly finite rings
1247:is Hausdorff (in the
1206:
1147:
641:
420:
226:
198:
175:if for every element
166:
78:absolutely flat rings
1493:semihereditary rings
1303:improve this section
1159:
1152:factors through the
1101:
1025:(the unique element
902:short exact sequence
760:affiliated operators
649:More generally, the
468:
357:
337:Gaussian elimination
211:
183:
151:
104:von Neumann algebras
1729:1936PNAS...22..707V
1501:semiprimitive rings
1154:morphism of schemes
812:is generated by an
764:von Neumann algebra
341:invertible matrices
133:if there exists an
120:commutative algebra
116:regular local rings
108:continuous geometry
60:. One may think of
1662:Journal of Algebra
1372:, there is a unit
1343:unit regular rings
1201:
1142:
820:finitely generated
814:idempotent element
636:
618:
566:
523:
415:
406:
221:
193:
161:
1823:von Neumann, John
1713:von Neumann, John
1627:Kaplansky, Irving
1513:Regular semigroup
1497:nonsingular rings
1437:subdirect product
1339:
1338:
1331:
735:endomorphism ring
703:is some ring and
680:endomorphism ring
670:over a field (or
1868:
1861:John von Neumann
1841:
1818:
1775:
1758:
1740:
1708:
1690:
1688:
1678:
1651:
1631:Fields and rings
1622:
1620:
1597:
1592:
1586:
1581:
1575:
1574:
1563:
1557:
1551:
1545:
1539:
1434:
1417:, there is some
1389:
1334:
1327:
1323:
1320:
1314:
1283:
1275:
1249:Zariski topology
1235:
1210:
1208:
1207:
1202:
1200:
1199:
1194:
1185:
1184:
1179:
1151:
1149:
1148:
1143:
1141:
1140:
1135:
1117:
1096:regular function
1093:
1082:
1048:
1038:
1024:
961:
951:
927:{0} and is thus
925:Jacobson radical
782:
658:
645:
643:
642:
637:
623:
622:
598:
597:
571:
570:
546:
545:
528:
527:
503:
502:
460:
424:
422:
421:
416:
411:
410:
386:
385:
334:
273:
263:
240:
230:
228:
227:
222:
220:
219:
202:
200:
199:
194:
192:
191:
170:
168:
167:
162:
160:
159:
146:
59:
42:there exists an
1876:
1875:
1871:
1870:
1869:
1867:
1866:
1865:
1846:
1845:
1844:
1800:
1783:
1781:Further reading
1778:
1723:(12): 707–713,
1641:
1606:
1601:
1600:
1593:
1589:
1582:
1578:
1568:
1566:Skornyakov 2001
1564:
1560:
1552:
1548:
1540:
1531:
1526:
1509:
1426:
1392:semisimple ring
1381:
1335:
1324:
1318:
1315:
1300:
1284:
1273:
1222:
1195:
1190:
1189:
1180:
1175:
1174:
1160:
1157:
1156:
1136:
1131:
1130:
1104:
1102:
1099:
1098:
1084:
1070:
1040:
1030:
1016:
988:Krull dimension
953:
943:
885:absolutely flat
789:
774:
749:semisimple ring
742:
687:
650:
617:
616:
611:
605:
604:
599:
593:
589:
582:
581:
565:
564:
559:
553:
552:
547:
541:
537:
530:
529:
522:
521:
516:
510:
509:
504:
498:
494:
487:
486:
469:
466:
465:
449:
446:identity matrix
436:
405:
404:
399:
393:
392:
387:
381:
377:
370:
369:
358:
355:
354:
328:
319:
304:square matrices
290:
276:integral domain
265:
258:
247:
232:
215:
214:
212:
209:
208:
187:
186:
184:
181:
180:
155:
154:
152:
149:
148:
138:
51:
17:
12:
11:
5:
1874:
1864:
1863:
1858:
1843:
1842:
1819:
1798:
1784:
1782:
1779:
1777:
1776:
1709:
1691:
1669:(1): 185–201.
1652:
1639:
1623:
1607:
1605:
1602:
1599:
1598:
1587:
1576:
1558:
1554:Kaplansky 1972
1546:
1542:Kaplansky 1972
1528:
1527:
1525:
1522:
1521:
1520:
1515:
1508:
1505:
1485:
1484:
1477:
1464:
1455:
1441:division rings
1337:
1336:
1287:
1285:
1278:
1272:
1269:
1268:
1267:
1252:
1237:
1213:
1212:
1198:
1193:
1188:
1183:
1178:
1173:
1170:
1167:
1164:
1139:
1134:
1129:
1126:
1123:
1120:
1116:
1113:
1110:
1107:
1083:determined by
1060:
1050:
1010:
995:
981:
914:
913:
898:
894:weak dimension
869:
847:
837:
827:direct summand
823:
816:
803:
788:
785:
738:
723:(such modules
683:
647:
646:
635:
632:
629:
626:
621:
615:
612:
610:
607:
606:
603:
600:
596:
592:
588:
587:
585:
580:
577:
574:
569:
563:
560:
558:
555:
554:
551:
548:
544:
540:
536:
535:
533:
526:
520:
517:
515:
512:
511:
508:
505:
501:
497:
493:
492:
490:
485:
482:
479:
476:
473:
432:
426:
425:
414:
409:
403:
400:
398:
395:
394:
391:
388:
384:
380:
376:
375:
373:
368:
365:
362:
324:
286:
280:direct product
246:
243:
218:
190:
158:
15:
9:
6:
4:
3:
2:
1873:
1862:
1859:
1857:
1854:
1853:
1851:
1840:
1836:
1832:
1828:
1824:
1820:
1817:
1813:
1809:
1805:
1801:
1799:0-89464-632-X
1795:
1791:
1786:
1785:
1774:
1770:
1766:
1762:
1757:
1752:
1748:
1744:
1739:
1734:
1730:
1726:
1722:
1718:
1714:
1710:
1707:
1703:
1702:
1697:
1692:
1687:
1682:
1677:
1672:
1668:
1664:
1663:
1658:
1653:
1650:
1646:
1642:
1640:0-226-42451-0
1636:
1632:
1628:
1624:
1619:
1614:
1609:
1608:
1596:
1591:
1585:
1580:
1572:
1567:
1562:
1556:, p. 112
1555:
1550:
1544:, p. 110
1543:
1538:
1536:
1534:
1529:
1519:
1516:
1514:
1511:
1510:
1504:
1502:
1498:
1495:, left/right
1494:
1490:
1482:
1478:
1476:
1472:
1468:
1465:
1463:
1459:
1456:
1453:
1450:
1449:
1448:
1446:
1442:
1438:
1433:
1429:
1424:
1420:
1416:
1412:
1409:if for every
1408:
1404:
1399:
1397:
1393:
1388:
1384:
1379:
1375:
1371:
1367:
1364:if for every
1363:
1359:
1354:
1352:
1348:
1344:
1333:
1330:
1322:
1312:
1308:
1304:
1298:
1297:
1293:
1288:This section
1286:
1282:
1277:
1276:
1265:
1263:
1257:
1253:
1250:
1246:
1242:
1238:
1233:
1229:
1225:
1221:
1220:
1219:
1218:
1196:
1181:
1171:
1165:
1155:
1137:
1121:
1097:
1091:
1087:
1081:
1077:
1073:
1068:
1064:
1061:
1058:
1054:
1051:
1047:
1043:
1037:
1033:
1028:
1023:
1019:
1014:
1011:
1008:
1007:maximal ideal
1004:
1000:
996:
993:
989:
985:
982:
979:
976:
975:
974:
972:
967:
965:
960:
956:
950:
946:
941:
937:
932:
930:
929:semiprimitive
926:
921:
919:
911:
907:
903:
899:
896:
895:
890:
886:
882:
878:
874:
870:
868:
864:
860:
856:
852:
848:
846:
842:
838:
836:
832:
828:
824:
821:
817:
815:
811:
808:
804:
801:
798:
797:
796:
794:
784:
781:
777:
772:
767:
765:
761:
756:
754:
750:
746:
741:
736:
732:
731:
726:
722:
718:
714:
710:
706:
702:
697:
695:
691:
686:
681:
677:
673:
669:
665:
660:
657:
653:
633:
630:
627:
624:
619:
613:
608:
601:
594:
590:
583:
578:
575:
572:
567:
561:
556:
549:
542:
538:
531:
524:
518:
513:
506:
499:
495:
488:
483:
480:
477:
474:
471:
464:
463:
462:
459:
456:
452:
448:). If we set
447:
444:
440:
435:
431:
412:
407:
401:
396:
389:
382:
378:
371:
366:
363:
360:
353:
352:
351:
349:
345:
342:
338:
332:
327:
322:
317:
313:
309:
305:
302:
298:
294:
289:
283:
281:
277:
272:
268:
261:
256:
252:
242:
239:
235:
206:
178:
174:
173:regular ideal
145:
141:
136:
132:
128:
123:
121:
117:
113:
112:regular rings
109:
105:
101:
97:
92:
90:
86:
84:
79:
75:
71:
67:
63:
58:
54:
49:
45:
41:
37:
33:
30:
26:
22:
1826:
1789:
1720:
1716:
1699:
1666:
1660:
1630:
1590:
1579:
1561:
1549:
1518:Weak inverse
1488:
1486:
1480:
1470:
1466:
1457:
1451:
1444:
1431:
1427:
1422:
1418:
1414:
1410:
1406:
1402:
1400:
1386:
1382:
1377:
1373:
1369:
1365:
1362:unit regular
1361:
1357:
1355:
1346:
1342:
1340:
1325:
1316:
1301:Please help
1289:
1261:
1244:
1231:
1227:
1223:
1216:
1214:
1089:
1085:
1079:
1075:
1071:
1062:
1052:
1045:
1041:
1035:
1031:
1026:
1021:
1017:
1012:
1002:
999:localization
983:
977:
970:
968:
963:
958:
954:
948:
944:
939:
935:
933:
922:
917:
915:
908:-modules is
905:
892:
888:
884:
880:
872:
866:
862:
858:
844:
840:
834:
830:
829:of the left
799:
792:
790:
779:
775:
771:Boolean ring
768:
762:of a finite
758:The ring of
757:
744:
739:
733:). Then the
728:
724:
720:
716:
708:
704:
700:
698:
693:
689:
684:
675:
668:vector space
663:
661:
655:
651:
648:
457:
454:
450:
442:
438:
433:
429:
427:
347:
343:
330:
325:
320:
311:
307:
300:
296:
292:
287:
284:
270:
266:
264:we can take
259:
248:
237:
233:
204:
176:
147:. An ideal
143:
139:
134:
130:
126:
124:
93:
82:
77:
73:
69:
65:
61:
56:
52:
47:
43:
39:
35:
31:
24:
18:
1856:Ring theory
1319:August 2023
1009:is a field.
875:-module is
871:every left
727:are called
678:, then the
253:(and every
125:An element
96:von Neumann
68:in general
21:mathematics
1850:Categories
1839:0171.28003
1816:0749.16001
1773:0015.38802
1747:62.1103.03
1649:1001.16500
1618:2207.09929
1604:References
1405:is called
1380:such that
1360:is called
1351:rank rings
1029:such that
942:such that
910:pure exact
855:projective
810:left ideal
753:Noetherian
730:semisimple
672:skew field
350:such that
255:skew field
231:such that
137:such that
1706:EMS Press
1390:. Every
1290:does not
1266:coincide.
1187:→
1172:⊔
1128:→
990:0 and is
851:submodule
807:principal
713:submodule
1825:(1960),
1765:16577757
1629:(1972),
1507:See also
1241:spectrum
1065:has the
904:of left
861:-module
843:-module
833:-module
245:Examples
1808:1150975
1756:1076849
1725:Bibcode
1475:central
1462:reduced
1401:A ring
1356:A ring
1311:removed
1296:sources
992:reduced
891:having
461:, then
437:is the
428:(where
314:is the
98: (
85:-module
1837:
1814:
1806:
1796:
1771:
1763:
1753:
1745:
1647:
1637:
1230:/ nil(
1057:V-ring
997:Every
900:every
883:being
818:every
805:every
707:is an
339:gives
310:. If
274:. An
249:Every
1613:arXiv
1524:Notes
1425:with
1260:Spec(
1055:is a
1005:at a
887:, or
857:left
853:of a
787:Facts
666:is a
295:) of
251:field
50:with
27:is a
1794:ISBN
1761:PMID
1635:ISBN
1571:help
1499:and
1349:and
1345:and
1294:any
1292:cite
1254:The
1239:The
1092:, 0)
1039:and
986:has
952:and
877:flat
441:-by-
346:and
316:rank
299:-by-
114:and
106:and
100:1936
89:flat
29:ring
23:, a
1835:Zbl
1812:Zbl
1769:Zbl
1751:PMC
1743:JFM
1733:doi
1681:hdl
1671:doi
1645:Zbl
1473:is
1439:of
1432:aax
1421:in
1413:in
1387:aua
1376:in
1368:in
1305:by
1243:of
1088:↦ (
1042:yxy
1032:xyx
1001:of
955:yxy
945:xyx
737:End
715:of
682:End
662:If
323:∈ M
318:of
262:≠ 0
238:axa
207:in
179:in
144:axa
118:of
87:is
57:axa
46:in
38:in
19:In
1852::
1833:,
1829:,
1810:,
1804:MR
1802:,
1767:,
1759:,
1749:,
1741:,
1731:,
1721:22
1719:,
1704:,
1698:,
1679:.
1667:25
1665:.
1659:.
1643:,
1532:^
1503:.
1447::
1430:=
1385:=
1353:.
1251:).
1226:=
1078:×
1074:→
1049:).
1044:=
1034:=
1020:∈
973::
966:.
795::
778:=
769:A
674:)
654:×
453:=
335:,
269:=
241:.
236:=
142:=
122:.
91:.
66:a;
55:=
1735::
1727::
1689:.
1683::
1673::
1621:.
1615::
1573:)
1489:π
1481:R
1471:R
1467:R
1458:R
1452:R
1445:R
1428:a
1423:R
1419:x
1415:R
1411:a
1403:R
1383:a
1378:R
1374:u
1370:R
1366:a
1358:R
1332:)
1326:(
1321:)
1317:(
1313:.
1299:.
1264:)
1262:A
1245:A
1234:)
1232:A
1228:A
1224:R
1217:A
1211:.
1197:1
1192:A
1182:m
1177:G
1169:}
1166:0
1163:{
1138:1
1133:A
1125:)
1122:R
1119:(
1115:c
1112:e
1109:p
1106:S
1090:t
1086:t
1080:Z
1076:Z
1072:Z
1063:R
1059:.
1053:R
1046:y
1036:x
1027:y
1022:R
1018:x
1013:R
1003:R
994:.
984:R
978:R
971:R
964:x
959:y
957:=
949:x
947:=
940:y
936:x
918:R
912:.
906:R
897:0
889:R
881:R
873:R
867:P
863:P
859:R
845:R
841:R
835:R
831:R
800:R
793:R
780:a
776:a
745:M
743:(
740:S
725:M
721:M
717:M
709:S
705:M
701:S
694:V
690:V
688:(
685:K
676:K
664:V
656:n
652:n
634:.
631:A
628:=
625:V
620:)
614:0
609:0
602:0
595:r
591:I
584:(
579:U
576:=
573:V
568:)
562:0
557:0
550:0
543:r
539:I
532:(
525:)
519:0
514:0
507:0
500:r
496:I
489:(
484:U
481:=
478:A
475:X
472:A
458:U
455:V
451:X
443:r
439:r
434:r
430:I
413:V
408:)
402:0
397:0
390:0
383:r
379:I
372:(
367:U
364:=
361:A
348:V
344:U
333:)
331:K
329:(
326:n
321:A
312:r
308:K
301:n
297:n
293:K
291:(
288:n
271:a
267:x
260:a
234:a
217:i
205:x
189:i
177:a
157:i
140:a
135:x
127:a
83:R
74:a
70:x
62:x
53:a
48:R
44:x
40:R
36:a
32:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.