25:
303:
1162:
1066:
1026:
595:
1538:
548:
511:
1628:
1567:
1504:
1482:
1127:
1098:
991:
965:
923:
257:
1797:
54:
700:
871:). This alternate definition gives a strictly weaker condition, even for rings that do have a multiplicative identity, in that all
157:
879:. With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of
1781:
1913:
1882:
1853:
1825:
1754:
693:
645:
76:
47:
267:
1132:
1036:
996:
562:
1509:
686:
553:
403:
1901:
1268:, including the additive identity ("empty combination") and multiplicative identity ("empty product").
163:
524:
487:
178:
37:
1074:
849:
Some mathematicians define rings without requiring the existence of a multiplicative identity (see
638:
441:
391:
41:
33:
1611:
1550:
1487:
1456:
1110:
1081:
974:
939:
906:
240:
1231:
745:
450:
184:
143:
1905:
851:
607:
458:
409:
190:
58:
834:
1841:
1932:
1744:
1210:
1206:
1202:
331:
205:
875:
become subrings, and they may have a multiplicative identity that differs from the one of
8:
1676:
1185:
926:
872:
613:
421:
372:
317:
211:
197:
125:
93:
1894:
1791:
1666:
1635:
1261:
722:
626:
112:
1909:
1878:
1849:
1821:
1777:
1750:
1451:
667:
464:
229:
170:
1605:
1194:
1170:
901:
737:
673:
659:
473:
415:
378:
151:
137:
1771:
1714:
1671:
1364:
1214:
934:
435:
385:
223:
1541:
826:
479:
1926:
1681:
620:
516:
131:
1198:
652:
427:
323:
1590:
1028:
have no subrings (with multiplicative identity) other than the full ring.
930:
714:
632:
343:
217:
99:
1218:
397:
837:
under multiplication and subtraction. This is sometimes known as the
807:
357:
262:
1848:. Washington, DC: Mathematical Association of America. p. 145.
1749:(Third ed.). Hoboken, NJ: John Wiley & Sons. p. 228.
1631:
795:
351:
337:
1877:. University Mathematical Texts. Oliver and Boyd. pp. 14–16.
1033:
Every ring has a unique smallest subring, isomorphic to some ring
1441:. Individual elements can also be adjoined to a subring, denoted
235:
729:
1248:, which is defined as the intersection of all subrings of
1713:
Not to be confused with the ring-theoretic analog of a
744:
are restricted to the subset, and that shares the same
1645:
is the smallest positive integer such that the sum of
867:(this does imply it contains the additive identity of
776:
that preserves the structure of the ring, i.e. a ring
1614:
1553:
1512:
1490:
1459:
1135:
1113:
1084:
1039:
999:
977:
942:
909:
565:
527:
490:
270:
243:
1743:
Dummit, David Steven; Foote, Richard Martin (2004).
1738:
1736:
1734:
1893:
1622:
1561:
1532:
1498:
1476:
1156:
1121:
1092:
1060:
1020:
985:
959:
917:
589:
542:
505:
297:
251:
1731:
1924:
46:but its sources remain unclear because it lacks
1818:Abstract Algebra: Structures and Applications
1225:
694:
1527:
1521:
298:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
1820:. Boca Raton: CRC Press. pp. 216–217.
1742:
1577:The intersection of all subrings of a ring
829:it contains the multiplicative identity of
1796:: CS1 maint: location missing publisher (
1157:{\displaystyle \mathbb {Z} /0\mathbb {Z} }
1061:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
1021:{\displaystyle \mathbb {Z} /n\mathbb {Z} }
701:
687:
1616:
1555:
1514:
1492:
1461:
1264:with integer coefficients of elements of
1150:
1137:
1115:
1086:
1054:
1041:
1014:
1001:
979:
944:
911:
590:{\displaystyle \mathbb {Z} (p^{\infty })}
567:
530:
493:
291:
278:
245:
77:Learn how and when to remove this message
16:Subset of a ring that forms a ring itself
1811:
1809:
1807:
1776:(3 ed.). New York. pp. 89–90.
1197:has subrings isomorphic to the rings of
1872:
1925:
1891:
1840:
1815:
1700:In general, not all subsets of a ring
1533:{\displaystyle \mathbb {Z} \cup \{i\}}
1425:is a ring extension and is said to be
1279:; therefore, the subring generated by
1866:
1804:
863:that is a ring for the operations of
1846:A Guide to Groups, Rings, and Fields
1769:
1581:is a subring that may be called the
1540:, and thus is the adjunction of the
1363:Subrings generalize some aspects of
1236:A special kind of subring of a ring
1217:, the subrings can be identified as
158:Free product of associative algebras
18:
13:
740:of addition and multiplication on
579:
14:
1944:
1816:Lovett, Stephen (2015). "Rings".
1358:
852:Ring (mathematics) § History
646:Noncommutative algebraic geometry
1572:
1271:Any intersection of subrings of
543:{\displaystyle \mathbb {Q} _{p}}
506:{\displaystyle \mathbb {Z} _{p}}
23:
1834:
1763:
1707:
1694:
1600:is a subring of the center of
1471:
1465:
954:
948:
855:). In this case, a subring of
794:. Equivalently, it is both a
584:
571:
1:
1844:(2012). "Rings and Modules".
1724:
1209:. Since these rings are also
844:
755:
1875:Elementary rings and modules
1623:{\displaystyle \mathbb {Z} }
1596:The prime subring of a ring
1562:{\displaystyle \mathbb {Z} }
1499:{\displaystyle \mathbb {C} }
1477:{\displaystyle \mathbb {Z} }
1390:
1122:{\displaystyle \mathbb {Z} }
1093:{\displaystyle \mathbb {Z} }
986:{\displaystyle \mathbb {Z} }
960:{\displaystyle \mathbb {Z} }
918:{\displaystyle \mathbb {Z} }
252:{\displaystyle \mathbb {Z} }
7:
1660:
1344:, it is said that the ring
1295:is the smallest subring of
1256:. The subring generated by
1072:a nonnegative integer (see
894:
736:that is itself a ring when
404:Unique factorization domain
10:
1949:
1902:Cambridge University Press
1229:
1226:Subring generated by a set
164:Tensor product of algebras
1873:Adamson, Iain T. (1972).
1450:For example, the ring of
1287:) is indeed a subring of
1107:in this statement, since
925:is a subring of both the
1687:
1307:is any other subring of
442:Formal power series ring
392:Integrally closed domain
32:This article includes a
1896:Rings and factorization
1371:is a subring of a ring
1332:itself is a subring of
1275:is itself a subring of
1260:is also the set of all
1232:Generator (mathematics)
746:multiplicative identity
451:Algebraic number theory
144:Total ring of fractions
61:more precise citations.
1892:Sharpe, David (1987).
1634:or to the ring of the
1624:
1563:
1534:
1500:
1478:
1158:
1123:
1094:
1062:
1022:
987:
961:
919:
608:Noncommutative algebra
591:
544:
507:
459:Algebraic number field
410:Principal ideal domain
299:
253:
191:Frobenius endomorphism
1625:
1564:
1535:
1501:
1479:
1203:split-complex numbers
1159:
1124:
1095:
1063:
1023:
988:
962:
920:
883:that is a subring of
592:
545:
508:
300:
254:
1770:Lang, Serge (2002).
1612:
1551:
1510:
1488:
1457:
1375:, then equivalently
1207:complex number field
1133:
1111:
1082:
1037:
997:
975:
940:
907:
760:A subring of a ring
614:Noncommutative rings
563:
525:
488:
332:Non-associative ring
268:
241:
198:Algebraic structures
1842:Gouvêa, Fernando Q.
1677:Algebraic extension
1608:either to the ring
1421:is a subring, then
1262:linear combinations
1186:associative algebra
373:Commutative algebra
212:Associative algebra
94:Algebraic structure
1867:General references
1667:Integral extension
1620:
1559:
1530:
1496:
1474:
1154:
1119:
1090:
1058:
1018:
993:and its quotients
983:
957:
915:
627:Semiprimitive ring
587:
540:
503:
311:Related structures
295:
249:
185:Inner automorphism
171:Ring homomorphisms
34:list of references
1452:Gaussian integers
1283:(denoted here as
1195:split-quaternions
1129:is isomorphic to
738:binary operations
711:
710:
668:Geometric algebra
379:Commutative rings
230:Category of rings
87:
86:
79:
1940:
1919:
1899:
1888:
1860:
1859:
1838:
1832:
1831:
1813:
1802:
1801:
1795:
1787:
1767:
1761:
1760:
1746:Abstract algebra
1740:
1718:
1711:
1705:
1703:
1698:
1656:
1652:
1648:
1644:
1639:
1636:integers modulo
1629:
1627:
1626:
1621:
1619:
1603:
1599:
1589:by analogy with
1588:
1580:
1568:
1566:
1565:
1560:
1558:
1546:
1539:
1537:
1536:
1531:
1517:
1505:
1503:
1502:
1497:
1495:
1484:is a subring of
1483:
1481:
1480:
1475:
1464:
1446:
1440:
1434:
1428:
1424:
1420:
1416:
1406:
1403:is a subring of
1402:
1398:
1386:
1379:is said to be a
1378:
1374:
1370:
1365:field extensions
1354:
1347:
1343:
1340:is generated by
1339:
1335:
1331:
1324:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1267:
1259:
1255:
1251:
1247:
1239:
1188:over its center.
1183:
1179:
1176:is a subring of
1175:
1171:center of a ring
1163:
1161:
1160:
1155:
1153:
1145:
1140:
1128:
1126:
1125:
1120:
1118:
1106:
1099:
1097:
1096:
1091:
1089:
1078:). The integers
1067:
1065:
1064:
1059:
1057:
1049:
1044:
1027:
1025:
1024:
1019:
1017:
1009:
1004:
992:
990:
989:
984:
982:
966:
964:
963:
958:
947:
924:
922:
921:
916:
914:
902:ring of integers
890:
886:
882:
878:
870:
866:
862:
858:
832:
824:
817:
805:
793:
783:
775:
771:
767:
751:
735:
727:
703:
696:
689:
674:Operator algebra
660:Clifford algebra
596:
594:
593:
588:
583:
582:
570:
549:
547:
546:
541:
539:
538:
533:
512:
510:
509:
504:
502:
501:
496:
474:Ring of integers
468:
465:Integers modulo
416:Euclidean domain
304:
302:
301:
296:
294:
286:
281:
258:
256:
255:
250:
248:
152:Product of rings
138:Fractional ideal
97:
89:
88:
82:
75:
71:
68:
62:
57:this article by
48:inline citations
27:
26:
19:
1948:
1947:
1943:
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1941:
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1922:
1916:
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1805:
1789:
1788:
1784:
1768:
1764:
1757:
1741:
1732:
1727:
1722:
1721:
1715:group extension
1712:
1708:
1701:
1699:
1695:
1690:
1672:Group extension
1663:
1654:
1650:
1646:
1642:
1637:
1615:
1613:
1610:
1609:
1601:
1597:
1586:
1578:
1575:
1554:
1552:
1549:
1548:
1544:
1513:
1511:
1508:
1507:
1491:
1489:
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1485:
1460:
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1455:
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1442:
1436:
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1408:
1404:
1400:
1396:
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1384:
1376:
1372:
1368:
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1352:
1345:
1341:
1337:
1333:
1329:
1316:
1312:
1308:
1304:
1300:
1296:
1292:
1291:. This subring
1288:
1284:
1280:
1276:
1272:
1265:
1257:
1253:
1249:
1245:
1240:is the subring
1237:
1234:
1228:
1215:square matrices
1213:represented by
1181:
1177:
1173:
1149:
1141:
1136:
1134:
1131:
1130:
1114:
1112:
1109:
1108:
1101:
1085:
1083:
1080:
1079:
1053:
1045:
1040:
1038:
1035:
1034:
1013:
1005:
1000:
998:
995:
994:
978:
976:
973:
972:
943:
941:
938:
937:
935:polynomial ring
910:
908:
905:
904:
897:
888:
884:
880:
876:
868:
864:
860:
859:is a subset of
856:
847:
830:
822:
811:
799:
785:
777:
773:
769:
761:
758:
749:
733:
725:
707:
678:
677:
610:
600:
599:
578:
574:
566:
564:
561:
560:
534:
529:
528:
526:
523:
522:
497:
492:
491:
489:
486:
485:
466:
436:Polynomial ring
386:Integral domain
375:
365:
364:
290:
282:
277:
269:
266:
265:
244:
242:
239:
238:
224:Involutive ring
109:
98:
92:
83:
72:
66:
63:
52:
38:related reading
28:
24:
17:
12:
11:
5:
1946:
1936:
1935:
1921:
1920:
1914:
1889:
1883:
1868:
1865:
1862:
1861:
1854:
1833:
1826:
1803:
1783:978-0387953854
1782:
1762:
1755:
1729:
1728:
1726:
1723:
1720:
1719:
1706:
1692:
1691:
1689:
1686:
1685:
1684:
1679:
1674:
1669:
1662:
1659:
1618:
1574:
1571:
1557:
1542:imaginary unit
1529:
1526:
1523:
1520:
1516:
1494:
1473:
1470:
1467:
1463:
1399:is a ring and
1392:
1389:
1381:ring extension
1360:
1359:Ring extension
1357:
1303:; that is, if
1227:
1224:
1223:
1222:
1190:
1189:
1166:
1165:
1152:
1148:
1144:
1139:
1117:
1100:correspond to
1088:
1075:Characteristic
1056:
1052:
1048:
1043:
1030:
1029:
1016:
1012:
1008:
1003:
981:
969:
968:
956:
953:
950:
946:
913:
896:
893:
846:
843:
827:if and only if
821:Equivalently,
757:
754:
709:
708:
706:
705:
698:
691:
683:
680:
679:
671:
670:
642:
641:
635:
629:
623:
611:
606:
605:
602:
601:
598:
597:
586:
581:
577:
573:
569:
550:
537:
532:
513:
500:
495:
483:-adic integers
476:
470:
461:
447:
446:
445:
444:
438:
432:
431:
430:
418:
412:
406:
400:
394:
376:
371:
370:
367:
366:
363:
362:
361:
360:
348:
347:
346:
340:
328:
327:
326:
308:
307:
306:
305:
293:
289:
285:
280:
276:
273:
259:
247:
226:
220:
214:
208:
194:
193:
187:
181:
167:
166:
160:
154:
148:
147:
146:
140:
128:
122:
110:
108:Basic concepts
107:
106:
103:
102:
85:
84:
42:external links
31:
29:
22:
15:
9:
6:
4:
3:
2:
1945:
1934:
1931:
1930:
1928:
1917:
1915:0-521-33718-6
1911:
1907:
1903:
1898:
1897:
1890:
1886:
1884:0-05-002192-3
1880:
1876:
1871:
1870:
1857:
1855:9780883853559
1851:
1847:
1843:
1837:
1829:
1827:9781482248913
1823:
1819:
1812:
1810:
1808:
1799:
1793:
1785:
1779:
1775:
1774:
1766:
1758:
1756:0-471-43334-9
1752:
1748:
1747:
1739:
1737:
1735:
1730:
1716:
1710:
1697:
1693:
1683:
1682:Ore extension
1680:
1678:
1675:
1673:
1670:
1668:
1665:
1664:
1658:
1640:
1633:
1607:
1594:
1592:
1584:
1583:prime subring
1573:Prime subring
1570:
1543:
1524:
1518:
1506:generated by
1468:
1453:
1448:
1445:
1439:
1431:
1415:
1411:
1407:generated by
1388:
1382:
1366:
1356:
1351:
1326:
1323:
1319:
1269:
1263:
1243:
1233:
1220:
1216:
1212:
1211:real algebras
1208:
1205:, and to the
1204:
1200:
1196:
1192:
1191:
1187:
1172:
1168:
1167:
1146:
1142:
1104:
1077:
1076:
1071:
1050:
1046:
1032:
1031:
1010:
1006:
971:
970:
951:
936:
932:
928:
903:
899:
898:
892:
874:
854:
853:
842:
840:
836:
828:
825:is a subring
819:
815:
809:
803:
797:
792:
788:
782:, +, *, 0, 1)
781:
766:, +, *, 0, 1)
765:
753:
747:
743:
739:
731:
724:
720:
716:
704:
699:
697:
692:
690:
685:
684:
682:
681:
676:
675:
669:
665:
664:
663:
662:
661:
656:
655:
654:
649:
648:
647:
640:
636:
634:
630:
628:
624:
622:
621:Division ring
618:
617:
616:
615:
609:
604:
603:
575:
559:
557:
551:
535:
521:
520:-adic numbers
519:
514:
498:
484:
482:
477:
475:
471:
469:
462:
460:
456:
455:
454:
453:
452:
443:
439:
437:
433:
429:
425:
424:
423:
419:
417:
413:
411:
407:
405:
401:
399:
395:
393:
389:
388:
387:
383:
382:
381:
380:
374:
369:
368:
359:
355:
354:
353:
349:
345:
341:
339:
335:
334:
333:
329:
325:
321:
320:
319:
315:
314:
313:
312:
287:
283:
274:
271:
264:
263:Terminal ring
260:
237:
233:
232:
231:
227:
225:
221:
219:
215:
213:
209:
207:
203:
202:
201:
200:
199:
192:
188:
186:
182:
180:
176:
175:
174:
173:
172:
165:
161:
159:
155:
153:
149:
145:
141:
139:
135:
134:
133:
132:Quotient ring
129:
127:
123:
121:
117:
116:
115:
114:
105:
104:
101:
96:→ Ring theory
95:
91:
90:
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1933:Ring theory
1904:. pp.
1604:, which is
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1219:subalgebras
715:mathematics
633:Simple ring
344:Jordan ring
218:Graded ring
100:Ring theory
59:introducing
1725:References
1704:are rings.
1649:copies of
1606:isomorphic
1435:, denoted
1230:See also:
845:Variations
756:Definition
639:Commutator
398:GCD domain
1792:cite book
1519:∪
1391:Adjoining
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1661:See also
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895:Examples
891:itself.
796:subgroup
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1773:Algebra
1653:equals
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554:Prüfer
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1906:15–17
1688:Notes
1367:. If
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