1145:
989:
952:
233:
814:
741:
776:
525:
478:
441:
1549:
are non-trivial, then the converse is false: the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the
1140:{\displaystyle \mathbf {Z} /p_{1}^{n_{1}}\mathbf {Z} \ \times \ \mathbf {Z} /p_{2}^{n_{2}}\mathbf {Z} \ \times \ \cdots \ \times \ \mathbf {Z} /p_{k}^{n_{k}}\mathbf {Z} .}
187:
630:
686:, one says colloquially that a ring is the product of some rings if it is isomorphic to the direct product of these rings. For example, the
859:
1800:
1775:
87:
1839:
850:
623:
575:
197:
1453:
1450:
781:
708:
746:
492:
1792:
616:
483:
333:
1767:
1457:
93:
454:
417:
108:
1326:
1151:
835:
687:
1682:
1461:
568:
371:
321:
170:
1471:
natural isomorphism, meaning that it doesn't matter in which order one forms the direct product.
664:
380:
114:
73:
1755:
1446:
537:
388:
339:
120:
1601:
1834:
1810:
261:
135:
1818:
8:
1482:
543:
351:
302:
247:
141:
127:
55:
23:
1760:
1639:
1236:
656:
556:
42:
1796:
1771:
1413:
1198:
672:
660:
597:
394:
159:
100:
1814:
1531:
980:
699:
603:
589:
403:
345:
308:
67:
1806:
1795:, vol. 211 (Revised third ed.), New York: Springer-Verlag, p. 91,
1616:
1405:
365:
315:
153:
1674:
1642:
668:
409:
663:
of the underlying sets of several rings (possibly an infinity), equipped with
1828:
1696:
1620:
703:
550:
446:
61:
1426:
965:
582:
357:
253:
1561:
846:
683:
644:
562:
273:
147:
29:
1784:
1624:
1358:
1195:
327:
1409:
287:
192:
1703:
is an element of the product whose coordinates are all zero except
947:{\displaystyle n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots \ p_{k}^{n_{k}},}
281:
267:
165:
49:
1720:
is an element of the product with all coordinates zero except
1695:
A product of two or more non-trivial rings always has nonzero
1468:
679:
1442:
fails to map 1 to 1 and hence is not a ring homomorphism.
1325:
This shows that the product of rings is an instance of
1645:
all of its components are units, i.e., if and only if
1416:(with identity): for example, when two or more of the
992:
862:
784:
749:
711:
495:
457:
420:
200:
173:
1759:
1404:, but this is incorrect from the point of view of
1139:
946:
808:
770:
735:
519:
472:
435:
227:
181:
1826:
1467:Direct products are commutative and associative
1460:. A coproduct in the category of algebras is a
624:
1336:is finite, the underlying additive group of
228:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
809:{\displaystyle \mathbb {Z} /n\mathbb {Z} .}
736:{\displaystyle \mathbb {Z} /mn\mathbb {Z} }
771:{\displaystyle \mathbb {Z} /m\mathbb {Z} }
631:
617:
1596:. However, the converse is not true when
799:
786:
764:
751:
729:
713:
520:{\displaystyle \mathbb {Z} (p^{\infty })}
497:
460:
423:
221:
208:
175:
16:Ring built from other rings (mathematics)
1754:
1611:form an ideal not contained in any such
1327:products in the sense of category theory
1827:
1186:is a product of rings, then for every
1783:
1745: = 0 in the product ring.
88:Free product of associative algebras
1619:gives that it is contained in some
13:
1370:. In this case, some authors call
1224: th coordinate. The product
1220:which projects the product on the
678:Since direct products are defined
509:
14:
1851:
1267:is a ring homomorphism for every
851:Fundamental theorem of arithmetic
576:Noncommutative algebraic geometry
1454:algebras over a commutative ring
1130:
1098:
1072:
1040:
1026:
994:
473:{\displaystyle \mathbb {Q} _{p}}
436:{\displaystyle \mathbb {Z} _{p}}
1685:of the groups of units of the
1600:is infinite. For example, the
1538:is of this form. However, if
1534:is true, i.e., every ideal of
1361:of the additive groups of the
1228:together with the projections
514:
501:
1:
1793:Graduate Texts in Mathematics
1748:
1374:the "direct sum of the rings
1157:
1408:, since it is usually not a
182:{\displaystyle \mathbb {Z} }
7:
1445:(A finite coproduct in the
845:is written as a product of
818:
334:Unique factorization domain
10:
1856:
1768:Cambridge University Press
1542:is infinite and the rings
1458:tensor product of algebras
94:Tensor product of algebras
1152:Chinese remainder theorem
688:Chinese remainder theorem
1840:Operations on structures
1462:free product of algebras
836:ring of integers modulo
823:An important example is
665:componentwise operations
372:Formal power series ring
322:Integrally closed domain
653:direct product of rings
381:Algebraic number theory
74:Total ring of fractions
1568:if all but one of the
1150:This follows from the
1141:
948:
810:
772:
737:
659:that is formed by the
538:Noncommutative algebra
521:
474:
437:
389:Algebraic number field
340:Principal ideal domain
229:
183:
121:Frobenius endomorphism
1142:
949:
811:
773:
738:
690:may be stated as: if
522:
475:
438:
230:
184:
1762:Noncommutative rings
1589:is a prime ideal in
1530:is finite, then the
1429:, the inclusion map
1275:, then there exists
990:
860:
782:
747:
709:
544:Noncommutative rings
493:
455:
418:
262:Non-associative ring
198:
171:
128:Algebraic structures
1357:coincides with the
1128:
1070:
1024:
940:
912:
890:
303:Commutative algebra
142:Associative algebra
24:Algebraic structure
1582:and the remaining
1279:ring homomorphism
1237:universal property
1235:has the following
1137:
1107:
1049:
1003:
944:
919:
891:
869:
806:
768:
743:is the product of
733:
557:Semiprimitive ring
517:
470:
433:
241:Related structures
225:
179:
115:Inner automorphism
101:Ring homomorphisms
1802:978-0-387-95385-4
1777:978-0-88385-039-8
1414:category of rings
1199:ring homomorphism
1096:
1090:
1084:
1078:
1038:
1032:
918:
673:category of rings
661:Cartesian product
641:
640:
598:Geometric algebra
309:Commutative rings
160:Category of rings
1847:
1821:
1780:
1766:(5th ed.),
1765:
1521:
1441:
1403:
1356:
1312:
1292:
1266:
1247:is any ring and
1219:
1185:
1146:
1144:
1143:
1138:
1133:
1127:
1126:
1125:
1115:
1106:
1101:
1094:
1088:
1082:
1076:
1075:
1069:
1068:
1067:
1057:
1048:
1043:
1036:
1030:
1029:
1023:
1022:
1021:
1011:
1002:
997:
953:
951:
950:
945:
939:
938:
937:
927:
916:
911:
910:
909:
899:
889:
888:
887:
877:
815:
813:
812:
807:
802:
794:
789:
777:
775:
774:
769:
767:
759:
754:
742:
740:
739:
734:
732:
721:
716:
700:coprime integers
697:
693:
649:product of rings
633:
626:
619:
604:Operator algebra
590:Clifford algebra
526:
524:
523:
518:
513:
512:
500:
479:
477:
476:
471:
469:
468:
463:
442:
440:
439:
434:
432:
431:
426:
404:Ring of integers
398:
395:Integers modulo
346:Euclidean domain
234:
232:
231:
226:
224:
216:
211:
188:
186:
185:
180:
178:
82:Product of rings
68:Fractional ideal
27:
19:
18:
1855:
1854:
1850:
1849:
1848:
1846:
1845:
1844:
1825:
1824:
1803:
1778:
1751:
1728:
1711:
1690:
1663:
1658:) is a unit in
1653:
1617:axiom of choice
1609:
1594:
1587:
1580:
1573:
1554:
1547:
1522:is an ideal of
1519:
1514:
1501:
1490:
1479:
1435:
1430:
1424:
1406:category theory
1402:
1394:
1384:
1382:
1369:
1355:
1347:
1337:
1310:
1301: ∘
1299:
1294:
1280:
1264:
1253:
1248:
1233:
1217:
1206:
1201:
1184:
1176:
1163:
1160:
1129:
1121:
1117:
1116:
1111:
1102:
1097:
1071:
1063:
1059:
1058:
1053:
1044:
1039:
1025:
1017:
1013:
1012:
1007:
998:
993:
991:
988:
987:
983:to the product
962:
933:
929:
928:
923:
905:
901:
900:
895:
883:
879:
878:
873:
861:
858:
857:
821:
798:
790:
785:
783:
780:
779:
763:
755:
750:
748:
745:
744:
728:
717:
712:
710:
707:
706:
695:
691:
637:
608:
607:
540:
530:
529:
508:
504:
496:
494:
491:
490:
464:
459:
458:
456:
453:
452:
427:
422:
421:
419:
416:
415:
396:
366:Polynomial ring
316:Integral domain
305:
295:
294:
220:
212:
207:
199:
196:
195:
174:
172:
169:
168:
154:Involutive ring
39:
28:
22:
17:
12:
11:
5:
1853:
1843:
1842:
1837:
1823:
1822:
1801:
1781:
1776:
1756:Herstein, I.N.
1750:
1747:
1724:
1707:
1688:
1675:group of units
1661:
1649:
1643:if and only if
1607:
1592:
1585:
1578:
1571:
1552:
1545:
1517:
1506:
1488:
1477:
1433:
1420:
1398:
1386:
1378:
1365:
1351:
1339:
1323:
1322:
1308:
1297:
1262:
1251:
1231:
1215:
1204:
1180:
1168:
1159:
1156:
1148:
1147:
1136:
1132:
1124:
1120:
1114:
1110:
1105:
1100:
1093:
1087:
1081:
1074:
1066:
1062:
1056:
1052:
1047:
1042:
1035:
1028:
1020:
1016:
1010:
1006:
1001:
996:
960:
955:
954:
943:
936:
932:
926:
922:
915:
908:
904:
898:
894:
886:
882:
876:
872:
868:
865:
820:
817:
805:
801:
797:
793:
788:
766:
762:
758:
753:
731:
727:
724:
720:
715:
669:direct product
639:
638:
636:
635:
628:
621:
613:
610:
609:
601:
600:
572:
571:
565:
559:
553:
541:
536:
535:
532:
531:
528:
527:
516:
511:
507:
503:
499:
480:
467:
462:
443:
430:
425:
413:-adic integers
406:
400:
391:
377:
376:
375:
374:
368:
362:
361:
360:
348:
342:
336:
330:
324:
306:
301:
300:
297:
296:
293:
292:
291:
290:
278:
277:
276:
270:
258:
257:
256:
238:
237:
236:
235:
223:
219:
215:
210:
206:
203:
189:
177:
156:
150:
144:
138:
124:
123:
117:
111:
97:
96:
90:
84:
78:
77:
76:
70:
58:
52:
40:
38:Basic concepts
37:
36:
33:
32:
15:
9:
6:
4:
3:
2:
1852:
1841:
1838:
1836:
1833:
1832:
1830:
1820:
1816:
1812:
1808:
1804:
1798:
1794:
1790:
1786:
1782:
1779:
1773:
1769:
1764:
1763:
1757:
1753:
1752:
1746:
1744:
1740:
1737: ≠
1736:
1732:
1727:
1723:
1719:
1715:
1710:
1706:
1702:
1698:
1697:zero divisors
1693:
1691:
1684:
1680:
1676:
1672:
1668:
1664:
1657:
1652:
1648:
1644:
1641:
1637:
1633:
1628:
1626:
1622:
1621:maximal ideal
1618:
1614:
1610:
1603:
1599:
1595:
1588:
1581:
1575:are equal to
1574:
1567:
1563:
1559:
1556:. The ideal
1555:
1548:
1541:
1537:
1533:
1529:
1525:
1520:
1513:
1509:
1504:
1499:
1495:
1491:
1484:
1480:
1472:
1470:
1465:
1463:
1459:
1455:
1452:
1448:
1443:
1440:
1436:
1428:
1423:
1419:
1415:
1411:
1407:
1401:
1397:
1393:
1389:
1381:
1377:
1373:
1368:
1364:
1360:
1354:
1350:
1346:
1342:
1335:
1330:
1328:
1320:
1316:
1311:
1304:
1300:
1291:
1287:
1283:
1278:
1277:precisely one
1274:
1270:
1265:
1258:
1254:
1246:
1242:
1241:
1240:
1238:
1234:
1227:
1223:
1218:
1211:
1207:
1200:
1197:
1193:
1189:
1183:
1179:
1175:
1171:
1166:
1155:
1153:
1134:
1122:
1118:
1112:
1108:
1103:
1091:
1085:
1079:
1064:
1060:
1054:
1050:
1045:
1033:
1018:
1014:
1008:
1004:
999:
986:
985:
984:
982:
979:is naturally
978:
975:
971:
967:
964:are distinct
963:
941:
934:
930:
924:
920:
913:
906:
902:
896:
892:
884:
880:
874:
870:
866:
863:
856:
855:
854:
852:
848:
844:
840:
839:
833:
830:
826:
816:
803:
795:
791:
760:
756:
725:
722:
718:
705:
704:quotient ring
701:
689:
685:
681:
676:
674:
670:
666:
662:
658:
654:
650:
646:
634:
629:
627:
622:
620:
615:
614:
612:
611:
606:
605:
599:
595:
594:
593:
592:
591:
586:
585:
584:
579:
578:
577:
570:
566:
564:
560:
558:
554:
552:
551:Division ring
548:
547:
546:
545:
539:
534:
533:
505:
489:
487:
481:
465:
451:
450:-adic numbers
449:
444:
428:
414:
412:
407:
405:
401:
399:
392:
390:
386:
385:
384:
383:
382:
373:
369:
367:
363:
359:
355:
354:
353:
349:
347:
343:
341:
337:
335:
331:
329:
325:
323:
319:
318:
317:
313:
312:
311:
310:
304:
299:
298:
289:
285:
284:
283:
279:
275:
271:
269:
265:
264:
263:
259:
255:
251:
250:
249:
245:
244:
243:
242:
217:
213:
204:
201:
194:
193:Terminal ring
190:
167:
163:
162:
161:
157:
155:
151:
149:
145:
143:
139:
137:
133:
132:
131:
130:
129:
122:
118:
116:
112:
110:
106:
105:
104:
103:
102:
95:
91:
89:
85:
83:
79:
75:
71:
69:
65:
64:
63:
62:Quotient ring
59:
57:
53:
51:
47:
46:
45:
44:
35:
34:
31:
26:→ Ring theory
25:
21:
20:
1788:
1761:
1742:
1738:
1734:
1730:
1725:
1721:
1717:
1713:
1708:
1704:
1700:
1694:
1686:
1678:
1670:
1666:
1659:
1655:
1650:
1646:
1635:
1631:
1629:
1612:
1605:
1597:
1590:
1583:
1576:
1569:
1565:
1557:
1550:
1543:
1539:
1535:
1527:
1523:
1515:
1511:
1507:
1502:
1497:
1493:
1486:
1475:
1473:
1466:
1444:
1438:
1431:
1421:
1417:
1399:
1395:
1391:
1387:
1383:" and write
1379:
1375:
1371:
1366:
1362:
1352:
1348:
1344:
1340:
1333:
1331:
1324:
1318:
1314:
1306:
1302:
1295:
1289:
1285:
1281:
1276:
1272:
1268:
1260:
1256:
1249:
1244:
1229:
1225:
1221:
1213:
1209:
1202:
1191:
1187:
1181:
1177:
1173:
1169:
1164:
1161:
1149:
976:
973:
969:
958:
956:
847:prime powers
842:
837:
831:
828:
824:
822:
677:
652:
648:
642:
602:
588:
587:
583:Free algebra
581:
580:
574:
573:
542:
485:
447:
410:
379:
378:
358:Finite field
307:
254:Finite field
240:
239:
166:Initial ring
126:
125:
99:
98:
81:
41:
1835:Ring theory
1785:Lang, Serge
1630:An element
1562:prime ideal
1451:commutative
684:isomorphism
645:mathematics
563:Simple ring
274:Jordan ring
148:Graded ring
30:Ring theory
1829:Categories
1819:0984.00001
1749:References
1665:for every
1625:a fortiori
1615:, but the
1602:direct sum
1359:direct sum
1313:for every
1293:such that
1196:surjective
1194:we have a
1158:Properties
981:isomorphic
957:where the
667:. It is a
569:Commutator
328:GCD domain
1758:(2005) ,
1623:which is
1492:for each
1410:coproduct
1092:×
1086:⋯
1080:×
1034:×
914:⋯
510:∞
288:Semifield
1787:(2002),
1733:) where
1729: (
1712: (
1654: (
1532:converse
1447:category
1425:are non-
1284: :
1255: :
1208: :
819:Examples
282:Semiring
268:Lie ring
50:Subrings
1811:1878556
1789:Algebra
1741:, then
1683:product
1681:is the
1627:prime.
1604:of the
1500:, then
1427:trivial
1412:in the
968:, then
671:in the
484:Prüfer
86:•
1817:
1809:
1799:
1774:
1716:) and
1673:. The
1526:. If
1481:is an
1095:
1089:
1083:
1077:
1037:
1031:
966:primes
917:
834:, the
702:, the
136:Module
109:Kernel
1699:: if
1638:is a
1560:is a
1483:ideal
1469:up to
1456:is a
1332:When
849:(see
841:. If
680:up to
655:is a
488:-ring
352:Field
248:Field
56:Ideal
43:Rings
1797:ISBN
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