141:, since every right ideal is finitely generated. It is also a right BĂ©zout ring since all finitely generated right ideals are principal. Indeed, it is clear that principal right ideal rings are exactly the rings which are both right BĂ©zout and right Noetherian.
1604:
is known to be a principal left ideal domain which is not right
Noetherian, and hence it cannot be a principal right ideal ring. This shows that even for domains principal left and principal right ideal rings are different.
1487:
Arguing as in
Example 3. above and using the Zariski-Samuel theorem, it is easy to check that Hungerford's theorem is equivalent to the statement that any special principal ring is the quotient of a discrete valuation ring.
789:
1088:
1195:
961:
704:
1333:
and has an extremely simple ideal structure: there are only finitely many ideals, each of which is a power of the maximal ideal. For this reason, special principal rings are examples of
480:
380:
1133:
583:
258:
200:
1472:
1400:
529:
1252:
883:
836:
1303:
436:
998:
1560:
1215:
285:
1046:
1602:
1503:
which is not just a product of fields is a noncommutative right and left principal ideal domain. Every right and left ideal is a direct summand of
1329:; in particular they are isomorphic to a finite direct product of principal Artinian local rings. A local Artinian principal ring is called a
1340:
The following result gives a complete classification of principal rings in terms of special principal rings and principal ideal domains.
17:
712:
125:(PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain.
1697:
1051:
1567:
1149:
888:
1667:
643:
382:. Without much more effort, it can be shown that right BĂ©zout rings are also closed under finite direct products.
450:
324:
1093:
534:
209:
151:
1742:
1429:
1357:
1662:, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, pp. xx+385,
488:
441:
1224:
1412:
The proof applies the
Chinese Remainder theorem to a minimal primary decomposition of the zero ideal.
841:
794:
707:
93:
1528:
1520:
1273:
609:
is again a principal ring. Similarly, any quotient of a principal ring is again a principal ring.
419:
964:
145:
385:
Principal right ideal rings and right BĂ©zout rings are also closed under quotients, that is, if
1484:
The proof of
Hungerford's theorem employs Cohen's structure theorems for complete local rings.
969:
121:
1545:
1200:
606:
1747:
1677:
1633:
1218:
263:
1707:
8:
1010:
398:
46:
1572:
39:
413:
108:. Left BĂ©zout rings are defined similarly. These conditions are studied in domains as
1693:
1663:
1703:
1645:
74:
109:
1726:
1673:
1497:
617:
138:
116:
69:.) When this is satisfied for both left and right ideals, such as the case when
66:
65:. (The right and left ideals of this form, generated by one element, are called
1736:
1718:
1714:
1539:
1334:
1326:
1650:
1563:
31:
1685:
1000:
and, being a quotient of a principal ring, is itself a principal ring.
1725:, Graduate Texts in Mathematics, vol. 28, 29, Berlin, New York:
1266:
has at least two elements, then the ring also has zero divisors. If
1255:
397:
is also principal right ideal ring. This follows readily from the
1409:
is either a principal ideal domain or a special principal ring.
1325:
The principal rings constructed in
Example 5. above are always
137:
is a principal right ideal ring, then it is certainly a right
1313:
principal: take the ideal generated by the finite subsets of
1197:
forms a commutative principal ideal ring with unity, where
1320:
784:{\displaystyle R/I\cong \prod _{i=1}^{n}R/P_{i}^{a_{i}}}
1415:
There is also the following result, due to
Hungerford:
1636:(1968), "On the structure of principal ideal rings",
1575:
1548:
1432:
1360:
1276:
1227:
1203:
1152:
1096:
1054:
1013:
972:
891:
844:
797:
715:
646:
537:
491:
453:
422:
327:
266:
212:
154:
144:
Principal right ideal rings are closed under finite
1083:{\displaystyle {\mathfrak {m}}=\langle x,y\rangle }
1692:(Third ed.), Reading, Mass.: Addison-Wesley,
1596:
1554:
1466:
1394:
1297:
1246:
1209:
1189:
1127:
1082:
1040:
992:
955:
877:
830:
783:
698:
577:
523:
474:
430:
404:All properties above have left analogues as well.
374:
279:
252:
194:
1531:are seen to be both right and left BĂ©zout rings.
1190:{\displaystyle ({\mathcal {P}}(X),\Delta ,\cap )}
1734:
1615:
956:{\displaystyle R_{P_{i}}/P_{i}^{a_{i}}R_{P_{i}}}
389:is a proper ideal of principal right ideal ring
699:{\displaystyle I=\prod _{i=1}^{n}P_{i}^{a_{i}}}
605:4. The localization of a principal ring at any
1713:
1077:
1065:
636:is a principal ring. Indeed, we may factor
1481:is a quotient of a principal ideal domain.
1632:
1491:
1135:. Then R is a finite local ring which is
475:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
1649:
468:
455:
424:
375:{\displaystyle (x_{1},\ldots ,x_{n})R=A}
115:A principal ideal ring which is also an
1562:is a ring endomorphism which is not an
1128:{\displaystyle R=A/{\mathfrak {m}}^{2}}
578:{\displaystyle R=\prod _{i=1}^{n}R_{i}}
407:
253:{\displaystyle A=\prod _{i=1}^{n}A_{i}}
195:{\displaystyle R=\prod _{i=1}^{n}R_{i}}
14:
1735:
1660:A first course in noncommutative rings
1321:Structure theory for commutative PIR's
301:are principal right ideal rings, then
27:Ring in which every ideal is principal
1467:{\displaystyle \prod _{i=1}^{n}R_{i}}
1395:{\displaystyle \prod _{i=1}^{n}R_{i}}
128:
1684:
1657:
1621:
1426:can be written as a direct product
1354:can be written as a direct product
1114:
1057:
589:is a principal ring if and only if
524:{\displaystyle R_{1},\ldots ,R_{n}}
24:
1230:
1204:
1175:
1158:
791:, so it suffices to see that each
25:
1759:
1247:{\displaystyle {\mathcal {P}}(X)}
36:principal right (left) ideal ring
878:{\displaystyle R/P_{i}^{a_{i}}}
831:{\displaystyle R/P_{i}^{a_{i}}}
321:, and then it can be seen that
1638:Pacific Journal of Mathematics
1591:
1579:
1292:
1283:
1241:
1235:
1184:
1169:
1163:
1153:
1035:
1023:
885:is isomorphic to the quotient
640:as a product of prime powers:
360:
328:
13:
1:
1608:
1298:{\displaystyle I=(\bigcup I)}
1527:. Paralleling this example,
598:is a principal ring for all
431:{\displaystyle \mathbb {Z} }
45:in which every right (left)
7:
1350:be a principal ring. Then
202:, then each right ideal of
10:
1764:
1683:Pages 86 & 146-155 of
1422:be a principal ring. Then
1418:Theorem (Hungerford): Let
1007:be a finite field and put
838:is a principal ring. But
18:Principal right ideal ring
1529:von Neumann regular rings
1309:is infinite, the ring is
993:{\displaystyle R_{P_{i}}}
708:Chinese Remainder Theorem
393:, then the quotient ring
1507:, and so is of the form
1219:set symmetric difference
1651:10.2140/pjm.1968.25.543
1555:{\displaystyle \sigma }
1492:Noncommutative examples
1210:{\displaystyle \Delta }
965:discrete valuation ring
1598:
1556:
1468:
1453:
1396:
1381:
1344:Zariski–Samuel theorem
1331:special principal ring
1299:
1248:
1211:
1191:
1146:be a finite set. Then
1129:
1084:
1042:
994:
957:
879:
832:
785:
750:
700:
673:
624:be a nonzero ideal of
579:
564:
525:
476:
432:
376:
281:
254:
239:
196:
181:
122:principal ideal domain
1599:
1557:
1469:
1433:
1397:
1361:
1300:
1249:
1212:
1192:
1130:
1085:
1043:
995:
958:
880:
833:
786:
730:
701:
653:
628:. Then the quotient
607:multiplicative subset
580:
544:
526:
477:
433:
377:
282:
280:{\displaystyle A_{i}}
255:
219:
197:
161:
1573:
1568:skew polynomial ring
1546:
1430:
1358:
1274:
1225:
1201:
1150:
1094:
1052:
1011:
970:
889:
842:
795:
713:
644:
535:
489:
451:
420:
408:Commutative examples
399:isomorphism theorems
325:
287:is a right ideal of
264:
210:
152:
100:are principal, then
83:principal ideal ring
1743:Commutative algebra
1723:Commutative algebra
1658:Lam, T. Y. (2001),
1041:{\displaystyle A=k}
935:
874:
827:
780:
695:
57:) for some element
1594:
1552:
1464:
1392:
1295:
1270:is an ideal, then
1244:
1207:
1187:
1125:
1080:
1038:
990:
953:
914:
875:
853:
828:
806:
781:
759:
696:
674:
575:
521:
472:
428:
372:
277:
250:
192:
129:General properties
94:finitely generated
1699:978-0-201-55540-0
1597:{\displaystyle D}
106:right BĂ©zout ring
16:(Redirected from
1755:
1729:
1710:
1680:
1654:
1653:
1625:
1619:
1603:
1601:
1600:
1595:
1561:
1559:
1558:
1553:
1473:
1471:
1470:
1465:
1463:
1462:
1452:
1447:
1401:
1399:
1398:
1393:
1391:
1390:
1380:
1375:
1304:
1302:
1301:
1296:
1253:
1251:
1250:
1245:
1234:
1233:
1216:
1214:
1213:
1208:
1196:
1194:
1193:
1188:
1162:
1161:
1134:
1132:
1131:
1126:
1124:
1123:
1118:
1117:
1110:
1089:
1087:
1086:
1081:
1061:
1060:
1047:
1045:
1044:
1039:
999:
997:
996:
991:
989:
988:
987:
986:
962:
960:
959:
954:
952:
951:
950:
949:
934:
933:
932:
922:
913:
908:
907:
906:
905:
884:
882:
881:
876:
873:
872:
871:
861:
852:
837:
835:
834:
829:
826:
825:
824:
814:
805:
790:
788:
787:
782:
779:
778:
777:
767:
758:
749:
744:
723:
705:
703:
702:
697:
694:
693:
692:
682:
672:
667:
584:
582:
581:
576:
574:
573:
563:
558:
530:
528:
527:
522:
520:
519:
501:
500:
481:
479:
478:
473:
471:
463:
458:
442:integers modulo
437:
435:
434:
429:
427:
414:ring of integers
381:
379:
378:
373:
359:
358:
340:
339:
286:
284:
283:
278:
276:
275:
259:
257:
256:
251:
249:
248:
238:
233:
201:
199:
198:
193:
191:
190:
180:
175:
119:is said to be a
96:right ideals of
81:can be called a
75:commutative ring
67:principal ideals
21:
1763:
1762:
1758:
1757:
1756:
1754:
1753:
1752:
1733:
1732:
1727:Springer-Verlag
1700:
1670:
1629:
1628:
1620:
1616:
1611:
1574:
1571:
1570:
1547:
1544:
1543:
1498:semisimple ring
1494:
1480:
1458:
1454:
1448:
1437:
1431:
1428:
1427:
1408:
1386:
1382:
1376:
1365:
1359:
1356:
1355:
1335:uniserial rings
1323:
1317:, for example.
1275:
1272:
1271:
1254:represents the
1229:
1228:
1226:
1223:
1222:
1202:
1199:
1198:
1157:
1156:
1151:
1148:
1147:
1119:
1113:
1112:
1111:
1106:
1095:
1092:
1091:
1056:
1055:
1053:
1050:
1049:
1012:
1009:
1008:
982:
978:
977:
973:
971:
968:
967:
945:
941:
940:
936:
928:
924:
923:
918:
909:
901:
897:
896:
892:
890:
887:
886:
867:
863:
862:
857:
848:
843:
840:
839:
820:
816:
815:
810:
801:
796:
793:
792:
773:
769:
768:
763:
754:
745:
734:
719:
714:
711:
710:
688:
684:
683:
678:
668:
657:
645:
642:
641:
618:Dedekind domain
597:
569:
565:
559:
548:
536:
533:
532:
515:
511:
496:
492:
490:
487:
486:
467:
459:
454:
452:
449:
448:
423:
421:
418:
417:
410:
354:
350:
335:
331:
326:
323:
322:
320:
314:
307:
300:
293:
271:
267:
265:
262:
261:
244:
240:
234:
223:
211:
208:
207:
206:is of the form
186:
182:
176:
165:
153:
150:
149:
146:direct products
139:Noetherian ring
131:
117:integral domain
49:is of the form
28:
23:
22:
15:
12:
11:
5:
1761:
1751:
1750:
1745:
1731:
1730:
1711:
1698:
1681:
1668:
1655:
1644:(3): 543–547,
1634:Hungerford, T.
1627:
1626:
1613:
1612:
1610:
1607:
1593:
1590:
1587:
1584:
1581:
1578:
1551:
1493:
1490:
1478:
1461:
1457:
1451:
1446:
1443:
1440:
1436:
1406:
1389:
1385:
1379:
1374:
1371:
1368:
1364:
1327:Artinian rings
1322:
1319:
1294:
1291:
1288:
1285:
1282:
1279:
1243:
1240:
1237:
1232:
1206:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1160:
1155:
1122:
1116:
1109:
1105:
1102:
1099:
1079:
1076:
1073:
1070:
1067:
1064:
1059:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
985:
981:
976:
948:
944:
939:
931:
927:
921:
917:
912:
904:
900:
895:
870:
866:
860:
856:
851:
847:
823:
819:
813:
809:
804:
800:
776:
772:
766:
762:
757:
753:
748:
743:
740:
737:
733:
729:
726:
722:
718:
691:
687:
681:
677:
671:
666:
663:
660:
656:
652:
649:
593:
572:
568:
562:
557:
554:
551:
547:
543:
540:
518:
514:
510:
507:
504:
499:
495:
470:
466:
462:
457:
426:
409:
406:
371:
368:
365:
362:
357:
353:
349:
346:
343:
338:
334:
330:
318:
312:
305:
298:
294:. If all the
291:
274:
270:
247:
243:
237:
232:
229:
226:
222:
218:
215:
189:
185:
179:
174:
171:
168:
164:
160:
157:
130:
127:
110:BĂ©zout domains
87:principal ring
26:
9:
6:
4:
3:
2:
1760:
1749:
1746:
1744:
1741:
1740:
1738:
1728:
1724:
1720:
1716:
1712:
1709:
1705:
1701:
1695:
1691:
1687:
1682:
1679:
1675:
1671:
1669:0-387-95183-0
1665:
1661:
1656:
1652:
1647:
1643:
1639:
1635:
1631:
1630:
1624:, p. 21.
1623:
1618:
1614:
1606:
1588:
1585:
1582:
1576:
1569:
1565:
1549:
1541:
1540:division ring
1537:
1532:
1530:
1526:
1522:
1518:
1514:
1510:
1506:
1502:
1499:
1489:
1485:
1482:
1477:
1474:, where each
1459:
1455:
1449:
1444:
1441:
1438:
1434:
1425:
1421:
1416:
1413:
1410:
1405:
1402:, where each
1387:
1383:
1377:
1372:
1369:
1366:
1362:
1353:
1349:
1345:
1341:
1338:
1336:
1332:
1328:
1318:
1316:
1312:
1308:
1305:. If instead
1289:
1286:
1280:
1277:
1269:
1265:
1261:
1257:
1238:
1220:
1181:
1178:
1172:
1166:
1145:
1140:
1138:
1120:
1107:
1103:
1100:
1097:
1074:
1071:
1068:
1062:
1032:
1029:
1026:
1020:
1017:
1014:
1006:
1001:
983:
979:
974:
966:
946:
942:
937:
929:
925:
919:
915:
910:
902:
898:
893:
868:
864:
858:
854:
849:
845:
821:
817:
811:
807:
802:
798:
774:
770:
764:
760:
755:
751:
746:
741:
738:
735:
731:
727:
724:
720:
716:
709:
706:, and by the
689:
685:
679:
675:
669:
664:
661:
658:
654:
650:
647:
639:
635:
631:
627:
623:
619:
615:
610:
608:
603:
601:
596:
592:
588:
570:
566:
560:
555:
552:
549:
545:
541:
538:
531:be rings and
516:
512:
508:
505:
502:
497:
493:
483:
464:
460:
446:
445:
438:
415:
405:
402:
400:
396:
392:
388:
383:
369:
366:
363:
355:
351:
347:
344:
341:
336:
332:
317:
311:
304:
297:
290:
272:
268:
260:, where each
245:
241:
235:
230:
227:
224:
220:
216:
213:
205:
187:
183:
177:
172:
169:
166:
162:
158:
155:
147:
142:
140:
136:
126:
124:
123:
118:
113:
111:
107:
103:
99:
95:
90:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
41:
37:
33:
19:
1722:
1689:
1659:
1641:
1637:
1617:
1564:automorphism
1535:
1533:
1524:
1516:
1512:
1508:
1504:
1500:
1495:
1486:
1483:
1475:
1423:
1419:
1417:
1414:
1411:
1403:
1351:
1347:
1343:
1342:
1339:
1330:
1324:
1314:
1310:
1306:
1267:
1263:
1259:
1143:
1141:
1136:
1004:
1002:
637:
633:
629:
625:
621:
613:
611:
604:
599:
594:
590:
586:
484:
443:
439:
411:
403:
394:
390:
386:
384:
315:
309:
302:
295:
288:
203:
143:
134:
132:
120:
114:
105:
104:is called a
101:
97:
92:If only the
91:
86:
85:, or simply
82:
78:
70:
62:
58:
54:
50:
42:
35:
29:
1748:Ring theory
1715:Zariski, O.
1686:Lang, Serge
1566:, then the
1217:represents
1139:principal.
401:for rings.
32:mathematics
1737:Categories
1719:Samuel, P.
1708:0848.13001
1609:References
1521:idempotent
1589:σ
1550:σ
1435:∏
1363:∏
1287:⋃
1205:Δ
1182:∩
1176:Δ
1078:⟩
1066:⟨
732:∏
728:≅
655:∏
546:∏
506:…
345:…
221:∏
163:∏
1721:(1975),
1688:(1993),
1622:Lam 2001
1256:powerset
585:. Then
1690:Algebra
1678:1838439
1142:7. Let
1003:6. Let
963:of the
612:5. Let
485:3. Let
440:2. The
412:1. The
1706:
1696:
1676:
1666:
1519:is an
1515:where
1496:Every
1346:: Let
148:. If
1538:is a
1262:. If
616:be a
73:is a
47:ideal
38:is a
1694:ISBN
1664:ISBN
1542:and
1221:and
1090:and
620:and
40:ring
34:, a
1704:Zbl
1646:doi
1534:If
1523:of
1511:or
1311:not
1258:of
1137:not
395:R/I
133:If
61:of
30:In
1739::
1717:;
1702:,
1674:MR
1672:,
1642:25
1640:,
1513:Re
1509:eR
1337:.
1048:,
602:.
482:.
447::
416::
112:.
89:.
77:,
55:Rx
51:xR
1648::
1592:]
1586:,
1583:x
1580:[
1577:D
1536:D
1525:R
1517:e
1505:R
1501:R
1479:i
1476:R
1460:i
1456:R
1450:n
1445:1
1442:=
1439:i
1424:R
1420:R
1407:i
1404:R
1388:i
1384:R
1378:n
1373:1
1370:=
1367:i
1352:R
1348:R
1315:X
1307:X
1293:)
1290:I
1284:(
1281:=
1278:I
1268:I
1264:X
1260:X
1242:)
1239:X
1236:(
1231:P
1185:)
1179:,
1173:,
1170:)
1167:X
1164:(
1159:P
1154:(
1144:X
1121:2
1115:m
1108:/
1104:A
1101:=
1098:R
1075:y
1072:,
1069:x
1063:=
1058:m
1036:]
1033:y
1030:,
1027:x
1024:[
1021:k
1018:=
1015:A
1005:k
984:i
980:P
975:R
947:i
943:P
938:R
930:i
926:a
920:i
916:P
911:/
903:i
899:P
894:R
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865:a
859:i
855:P
850:/
846:R
822:i
818:a
812:i
808:P
803:/
799:R
775:i
771:a
765:i
761:P
756:/
752:R
747:n
742:1
739:=
736:i
725:I
721:/
717:R
690:i
686:a
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670:n
665:1
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651:=
648:I
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632:/
630:R
626:R
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614:R
600:i
595:i
591:R
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571:i
567:R
561:n
556:1
553:=
550:i
542:=
539:R
517:n
513:R
509:,
503:,
498:1
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469:Z
465:n
461:/
456:Z
444:n
425:Z
391:R
387:I
370:A
367:=
364:R
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356:n
352:x
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337:1
333:x
329:(
319:i
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313:i
310:x
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306:i
303:A
299:i
296:R
292:i
289:R
273:i
269:A
246:i
242:A
236:n
231:1
228:=
225:i
217:=
214:A
204:R
188:i
184:R
178:n
173:1
170:=
167:i
159:=
156:R
135:R
102:R
98:R
79:R
71:R
63:R
59:x
53:(
43:R
20:)
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