2358:. Isomorphic rings differ only by a relabeling of elements. Example: Up to isomorphism, there are four rings of order 4. (This means that there are four pairwise non-isomorphic rings of order 4 such that every other ring of order 4 is isomorphic to one of them.) On the other hand, up to isomorphism, there are eleven
2535:
Some authors use the term "ring" to refer to structures that do not require a multiplicative identity; instead of "rng", "ring", and "rng homomorphism", they use the terms "ring", "ring with identity", and "ring homomorphism", respectively. Because of this, some other authors, to avoid ambiguity,
891:
725:
244:
536:
720:
489:
452:
198:
708:
920:
2334:
is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism. One can prove that a ring homomorphism is an isomorphism if and only if it is
943:
641:
2771:
98:
2745:
2721:
2687:
634:
586:
17:
208:
886:{\displaystyle {\begin{aligned}f(a+b)&=f(a)+f(b),\\f(ab)&=f(a)f(b),\\f(1_{R})&=1_{S},\end{aligned}}}
503:
2763:
2675:
627:
494:
1045:). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.
344:
104:
465:
428:
119:
1757:
in the category of rings; in particular, the zero ring is not a zero object in the category of rings.
1256:
579:
382:
332:
181:
711:
391:
125:
84:
1875:
1732:
1034:
663:
548:
399:
350:
131:
681:
2338:
as a function on the underlying sets. If there exists a ring isomorphism between two rings
2798:
2781:
2697:
2648:
2136:
1030:
947:
These conditions imply that additive inverses and the additive identity are preserved too.
272:
146:
8:
2148:
2140:
1831:
1681:
Therefore, the class of all rings together with ring homomorphisms forms a category, the
1395:
1362:
1177:
991:. From the standpoint of ring theory, isomorphic rings have exactly the same properties.
899:
554:
362:
313:
258:
152:
138:
66:
34:
925:
2803:
2705:
2504:
2503:
is a ring epimorphism, but not a surjection. However, they are exactly the same as the
1824:
1158:
1116:
667:
567:
53:
2735:
2536:
explicitly specify that rings are unital and that homomorphisms preserve the identity.
2767:
2741:
2717:
2683:
2312:
1682:
1042:
608:
405:
170:
1033:
of two ring homomorphisms is a ring homomorphism. It follows that the rings forms a
2655:
2636:
1196:
961:
614:
600:
414:
356:
319:
92:
78:
2777:
2731:
2713:
2693:
2679:
2644:
2516:
1747:
1539:
1411:
1234:
376:
326:
164:
2667:
2632:
1944:
1728:
1067:
be a ring homomorphism. Then, directly from these definitions, one can deduce:
420:
2288:
is not the zero ring), since it does not map the multiplicative identity 1 of
1026:. A rng homomorphism between (unital) rings need not be a ring homomorphism.
2792:
2620:
2359:
1576:
1496:
1003:
561:
457:
72:
1753:
As the initial object is not isomorphic to the terminal object, there is no
2379:
593:
368:
264:
2490:
2039:
1933:
1754:
1459:
655:
573:
284:
158:
40:
2643:, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.,
2755:
1925:
1813:
1366:
1249:
338:
1917:). On the other hand, the zero function is always a rng homomorphism.
2766:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
2335:
1902:
1707:
1374:
957:
298:
203:
1843:
is a ring homomorphism (this is an example of a ring automorphism).
1038:
292:
278:
2489:
However, surjective ring homomorphisms are vastly different from
2317:
1299:
1270:. This can sometimes be used to show that between certain rings
1263:
176:
60:
2109:
is a ring homomorphism. More generally, given an abelian group
2396:
is a monomorphism that is not injective, then it sends some
2284:,0) is a rng homomorphism, but not a ring homomorphism (if
2200:
is a rng homomorphism (and rng endomorphism), with kernel 3
1248:
An homomorphism is injective if and only if kernel is the
2672:
Commutative algebra with a view toward algebraic geometry
2597:
2573:
2493:
in the category of rings. For example, the inclusion
1010:, defined as above except without the third condition
2561:
1746:
to the zero ring. This says that the zero ring is a
928:
902:
723:
684:
506:
468:
431:
211:
184:
2549:
1992:) is a surjective ring homomorphism. The kernel of
937:
914:
885:
702:
678:are rings, then a ring homomorphism is a function
530:
483:
446:
238:
192:
2585:
2373:
2790:
2631:
27:Structure-preserving function between two rings
2378:Injective ring homomorphisms are identical to
2318:Endomorphisms, isomorphisms, and automorphisms
2327:is a ring homomorphism from a ring to itself.
1006:, then the corresponding notion is that of a
635:
2369:is a ring isomorphism from a ring to itself.
2121:is equivalent to giving a ring homomorphism
1727:. This says that the ring of integers is an
1161:from the (multiplicative) group of units of
969:is also a ring homomorphism. In this case,
710:that preserves addition, multiplication and
239:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
1742:, there is a unique ring homomorphism from
2704:
2603:
1165:to the (multiplicative) group of units of
642:
628:
2026:is a ring homomorphism between the rings
531:{\displaystyle \mathbb {Z} (p^{\infty })}
508:
471:
434:
232:
219:
186:
2730:
2666:
2654:
2627:. Englewood Cliffs, N.J.: Prentice Hall.
2579:
2567:
2292:to the multiplicative identity (1,1) of
2038:induces a ring homomorphism between the
1245:is the kernel of some ring homomorphism.
14:
2791:
2486:is a monomorphism this is impossible.
1897:is a ring homomorphism if and only if
1717:, there is a unique ring homomorphism
1710:(the ring whose only element is zero).
1631:The composition of ring homomorphisms
2619:
2555:
1313:is the smallest subring contained in
2754:
2591:
2306:
2147:is a ring homomorphism that is also
1702:to 0 is a ring homomorphism only if
99:Free product of associative algebras
2641:Introduction to commutative algebra
24:
1889:are rings, the zero function from
1874:is a ring endomorphism called the
1241:. Every two-sided ideal in a ring
520:
25:
2815:
587:Noncommutative algebraic geometry
484:{\displaystyle \mathbb {Q} _{p}}
447:{\displaystyle \mathbb {Z} _{p}}
2158:
2072:be a vector space over a field
1996:consists of all polynomials in
1980:(substitute the imaginary unit
1317:, then every ring homomorphism
2529:
2374:Monomorphisms and epimorphisms
2227:There is no ring homomorphism
1816:ring homomorphism with kernel
1619:) is prime (maximal) ideal in
1134:) is a unit element such that
856:
843:
830:
824:
818:
812:
799:
790:
777:
771:
762:
756:
743:
731:
694:
525:
512:
13:
1:
2764:Graduate Texts in Mathematics
2676:Graduate Texts in Mathematics
2612:
2382:in the category of rings: If
1048:
2543:
1905:(otherwise it fails to map 1
1698:that sends every element of
1596:is prime (maximal) ideal in
1331:induces a ring homomorphism
193:{\displaystyle \mathbb {Z} }
7:
2710:Algebras, rings and modules
2678:. Vol. 150. New York:
2510:
1761:
1037:with ring homomorphisms as
345:Unique factorization domain
10:
2820:
2310:
662:is a structure-preserving
105:Tensor product of algebras
2480:are identical, but since
2414:. Consider the two maps
2262:are rings, the inclusion
1932:with coefficients in the
1750:in the category of rings.
2522:
2216:(which is isomorphic to
2143:over a commutative ring
2113:, a module structure on
1924:denotes the ring of all
1850:of prime characteristic
1571:is surjective, then ker(
1519:) is a maximal ideal of
703:{\displaystyle f:R\to S}
383:Formal power series ring
333:Integrally closed domain
2660:Algebra I, Chapters 1β3
2410:to the same element of
1678:is a ring homomorphism.
1651:is a ring homomorphism
1278:, no ring homomorphism
712:multiplicative identity
392:Algebraic number theory
85:Total ring of fractions
2000:that are divisible by
1876:Frobenius endomorphism
1546:) is a prime ideal of
1476:) is a prime ideal of
1266:the characteristic of
939:
916:
887:
704:
670:. More explicitly, if
549:Noncommutative algebra
532:
485:
448:
400:Algebraic number field
351:Principal ideal domain
240:
194:
132:Frobenius endomorphism
1365:(or more generally a
940:
917:
888:
705:
533:
486:
449:
241:
195:
2141:associative algebras
2137:algebra homomorphism
1947:, then the function
1534:are commutative and
1509:is surjective, then
1454:are commutative and
926:
900:
721:
682:
555:Noncommutative rings
504:
466:
429:
273:Non-associative ring
209:
182:
139:Algebraic structures
2706:Hazewinkel, Michiel
2505:strong epimorphisms
1832:complex conjugation
1668:, the identity map
1410:can be viewed as a
1402:) is a subfield of
1188:), is a subring of
915:{\displaystyle a,b}
314:Commutative algebra
153:Associative algebra
35:Algebraic structure
2633:Atiyah, Michael F.
1988:in the polynomial
1825:modular arithmetic
1159:group homomorphism
938:{\displaystyle R.}
935:
912:
883:
881:
700:
568:Semiprimitive ring
528:
481:
444:
252:Related structures
236:
190:
126:Inner automorphism
112:Ring homomorphisms
2773:978-0-387-95385-4
2637:Macdonald, Ian G.
2367:ring automorphism
2325:ring endomorphism
2313:Category of rings
2307:Category of rings
1984:for the variable
1683:category of rings
1561:are commutative,
1491:are commutative,
1439:) is an ideal of
1153:. In particular,
1043:Category of rings
660:ring homomorphism
652:
651:
609:Geometric algebra
320:Commutative rings
171:Category of rings
16:(Redirected from
2811:
2784:
2751:
2740:(2nd ed.).
2732:Jacobson, Nathan
2727:
2701:
2663:
2651:
2628:
2607:
2601:
2595:
2589:
2583:
2577:
2571:
2565:
2559:
2553:
2537:
2533:
2502:
2485:
2479:
2466:
2454:, respectively;
2395:
2332:ring isomorphism
2301:
2276:that sends each
2275:
2250:
2243:
2199:
2185:
2131:
2108:
2090:
2064:
2025:
2008:
2006:
1979:
1960:
1928:in the variable
1873:
1872:
1842:
1811:
1787:
1726:
1697:
1677:
1660:
1650:
1640:
1610:
1591:
1570:
1565:is a field, and
1514:
1508:
1471:
1434:
1382:
1353:
1330:
1298:is the smallest
1287:
1232:
1152:
1066:
1008:rng homomorphism
979:, and the rings
977:ring isomorphism
974:
968:
955:
944:
942:
941:
936:
921:
919:
918:
913:
892:
890:
889:
884:
882:
875:
874:
855:
854:
709:
707:
706:
701:
644:
637:
630:
615:Operator algebra
601:Clifford algebra
537:
535:
534:
529:
524:
523:
511:
490:
488:
487:
482:
480:
479:
474:
453:
451:
450:
445:
443:
442:
437:
415:Ring of integers
409:
406:Integers modulo
357:Euclidean domain
245:
243:
242:
237:
235:
227:
222:
199:
197:
196:
191:
189:
93:Product of rings
79:Fractional ideal
38:
30:
29:
21:
18:Ring isomorphism
2819:
2818:
2814:
2813:
2812:
2810:
2809:
2808:
2789:
2788:
2787:
2774:
2748:
2737:Basic algebra I
2724:
2714:Springer-Verlag
2690:
2680:Springer-Verlag
2668:Eisenbud, David
2615:
2610:
2604:Hazewinkel 2004
2602:
2598:
2590:
2586:
2578:
2574:
2566:
2562:
2554:
2550:
2546:
2541:
2540:
2534:
2530:
2525:
2517:Change of rings
2513:
2494:
2481:
2478:
2468:
2465:
2455:
2453:
2446:
2427:
2420:
2409:
2402:
2383:
2376:
2320:
2315:
2309:
2293:
2263:
2245:
2228:
2198:
2194:
2187:
2165:
2161:
2139:between unital
2122:
2092:
2077:
2076:. Then the map
2058:
2048:
2042:
2013:
2002:
2001:
1962:
1948:
1945:complex numbers
1916:
1910:
1868:
1855:
1834:
1802:
1789:
1768:
1764:
1748:terminal object
1738:For every ring
1718:
1713:For every ring
1689:
1669:
1652:
1642:
1632:
1601:
1592:is surjective,
1587:
1566:
1540:integral domain
1510:
1504:
1467:
1430:
1425:is an ideal of
1412:field extension
1378:
1351:
1344:
1337:
1332:
1318:
1311:
1296:
1279:
1235:two-sided ideal
1230:
1204:
1135:
1085:
1079:
1054:
1051:
1025:
1019:
970:
964:
951:
950:If in addition
927:
924:
923:
901:
898:
897:
880:
879:
870:
866:
859:
850:
846:
837:
836:
802:
784:
783:
746:
724:
722:
719:
718:
683:
680:
679:
648:
619:
618:
551:
541:
540:
519:
515:
507:
505:
502:
501:
475:
470:
469:
467:
464:
463:
438:
433:
432:
430:
427:
426:
407:
377:Polynomial ring
327:Integral domain
316:
306:
305:
231:
223:
218:
210:
207:
206:
185:
183:
180:
179:
165:Involutive ring
50:
39:
33:
28:
23:
22:
15:
12:
11:
5:
2817:
2807:
2806:
2801:
2786:
2785:
2772:
2752:
2746:
2728:
2722:
2702:
2688:
2664:
2652:
2629:
2621:Artin, Michael
2616:
2614:
2611:
2609:
2608:
2596:
2584:
2572:
2560:
2547:
2545:
2542:
2539:
2538:
2527:
2526:
2524:
2521:
2520:
2519:
2512:
2509:
2476:
2463:
2451:
2444:
2425:
2418:
2407:
2400:
2375:
2372:
2371:
2370:
2363:
2328:
2319:
2316:
2311:Main article:
2308:
2305:
2304:
2303:
2252:
2225:
2196:
2192:
2160:
2157:
2156:
2155:
2133:
2066:
2054:
2044:
2010:
1918:
1912:
1906:
1879:
1844:
1828:
1798:
1763:
1760:
1759:
1758:
1751:
1736:
1729:initial object
1711:
1686:
1679:
1664:For each ring
1662:
1625:
1624:
1584:
1551:
1524:
1481:
1444:
1419:
1384:
1355:
1349:
1342:
1335:
1309:
1294:
1289:
1257:characteristic
1253:
1246:
1226:
1193:
1174:
1113:
1087:
1081:
1075:
1050:
1047:
1021:
1015:
934:
931:
911:
908:
905:
894:
893:
878:
873:
869:
865:
862:
860:
858:
853:
849:
845:
842:
839:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
803:
801:
798:
795:
792:
789:
786:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
747:
745:
742:
739:
736:
733:
730:
727:
726:
699:
696:
693:
690:
687:
650:
649:
647:
646:
639:
632:
624:
621:
620:
612:
611:
583:
582:
576:
570:
564:
552:
547:
546:
543:
542:
539:
538:
527:
522:
518:
514:
510:
491:
478:
473:
454:
441:
436:
424:-adic integers
417:
411:
402:
388:
387:
386:
385:
379:
373:
372:
371:
359:
353:
347:
341:
335:
317:
312:
311:
308:
307:
304:
303:
302:
301:
289:
288:
287:
281:
269:
268:
267:
249:
248:
247:
246:
234:
230:
226:
221:
217:
214:
200:
188:
167:
161:
155:
149:
135:
134:
128:
122:
108:
107:
101:
95:
89:
88:
87:
81:
69:
63:
51:
49:Basic concepts
48:
47:
44:
43:
26:
9:
6:
4:
3:
2:
2816:
2805:
2802:
2800:
2797:
2796:
2794:
2783:
2779:
2775:
2769:
2765:
2761:
2757:
2753:
2749:
2747:9780486471891
2743:
2739:
2738:
2733:
2729:
2725:
2723:1-4020-2690-0
2719:
2715:
2711:
2707:
2703:
2699:
2695:
2691:
2689:0-387-94268-8
2685:
2681:
2677:
2673:
2669:
2665:
2661:
2657:
2653:
2650:
2646:
2642:
2638:
2634:
2630:
2626:
2622:
2618:
2617:
2605:
2600:
2593:
2588:
2582:, p. 103
2581:
2580:Jacobson 1985
2576:
2569:
2568:Eisenbud 1995
2564:
2558:, p. 353
2557:
2552:
2548:
2532:
2528:
2518:
2515:
2514:
2508:
2506:
2501:
2497:
2492:
2487:
2484:
2475:
2471:
2462:
2458:
2450:
2443:
2439:
2435:
2431:
2424:
2417:
2413:
2406:
2399:
2394:
2390:
2386:
2381:
2380:monomorphisms
2368:
2364:
2361:
2357:
2353:
2349:
2345:
2341:
2337:
2333:
2329:
2326:
2322:
2321:
2314:
2300:
2296:
2291:
2287:
2283:
2279:
2274:
2270:
2266:
2261:
2257:
2253:
2248:
2242:
2238:
2235:
2231:
2226:
2223:
2219:
2215:
2211:
2207:
2203:
2190:
2184:
2180:
2176:
2172:
2168:
2164:The function
2163:
2162:
2153:
2151:
2146:
2142:
2138:
2134:
2129:
2125:
2120:
2116:
2112:
2107:
2103:
2099:
2095:
2088:
2084:
2080:
2075:
2071:
2067:
2062:
2057:
2052:
2047:
2041:
2037:
2033:
2029:
2024:
2020:
2016:
2011:
2005:
1999:
1995:
1991:
1987:
1983:
1977:
1973:
1969:
1965:
1959:
1955:
1951:
1946:
1942:
1938:
1935:
1931:
1927:
1923:
1919:
1915:
1909:
1904:
1900:
1896:
1892:
1888:
1884:
1880:
1877:
1871:
1866:
1862:
1858:
1853:
1849:
1845:
1841:
1837:
1833:
1829:
1826:
1822:
1819:
1815:
1810:
1806:
1801:
1796:
1792:
1788:, defined by
1786:
1783:
1779:
1775:
1771:
1767:The function
1766:
1765:
1756:
1752:
1749:
1745:
1741:
1737:
1734:
1730:
1725:
1721:
1716:
1712:
1709:
1705:
1701:
1696:
1692:
1688:The zero map
1687:
1684:
1680:
1676:
1672:
1667:
1663:
1659:
1655:
1649:
1645:
1639:
1635:
1630:
1629:
1628:
1622:
1618:
1614:
1609:
1605:
1599:
1595:
1590:
1585:
1582:
1578:
1577:maximal ideal
1574:
1569:
1564:
1560:
1556:
1552:
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1522:
1518:
1513:
1507:
1502:
1498:
1497:maximal ideal
1494:
1490:
1486:
1482:
1479:
1475:
1470:
1465:
1461:
1457:
1453:
1449:
1445:
1442:
1438:
1433:
1428:
1424:
1420:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1383:is injective.
1381:
1376:
1372:
1368:
1364:
1360:
1356:
1352:
1345:
1338:
1329:
1325:
1321:
1316:
1312:
1305:
1302:contained in
1301:
1297:
1290:
1286:
1282:
1277:
1273:
1269:
1265:
1262:
1258:
1254:
1251:
1247:
1244:
1240:
1236:
1229:
1224:
1220:
1216:
1212:
1208:
1203:, defined as
1202:
1198:
1194:
1191:
1187:
1184:, denoted im(
1183:
1179:
1175:
1172:
1168:
1164:
1160:
1156:
1150:
1146:
1142:
1138:
1133:
1129:
1125:
1121:
1118:
1114:
1111:
1107:
1103:
1099:
1095:
1091:
1088:
1084:
1078:
1073:
1070:
1069:
1068:
1065:
1061:
1057:
1046:
1044:
1040:
1036:
1032:
1027:
1024:
1018:
1013:
1009:
1005:
1001:
997:
992:
990:
986:
982:
978:
973:
967:
963:
959:
954:
948:
945:
932:
929:
909:
906:
903:
876:
871:
867:
863:
861:
851:
847:
840:
833:
827:
821:
815:
809:
806:
804:
796:
793:
787:
780:
774:
768:
765:
759:
753:
750:
748:
740:
737:
734:
728:
717:
716:
715:
713:
697:
691:
688:
685:
677:
673:
669:
665:
661:
657:
645:
640:
638:
633:
631:
626:
625:
623:
622:
617:
616:
610:
606:
605:
604:
603:
602:
597:
596:
595:
590:
589:
588:
581:
577:
575:
571:
569:
565:
563:
562:Division ring
559:
558:
557:
556:
550:
545:
544:
516:
500:
498:
492:
476:
462:
461:-adic numbers
460:
455:
439:
425:
423:
418:
416:
412:
410:
403:
401:
397:
396:
395:
394:
393:
384:
380:
378:
374:
370:
366:
365:
364:
360:
358:
354:
352:
348:
346:
342:
340:
336:
334:
330:
329:
328:
324:
323:
322:
321:
315:
310:
309:
300:
296:
295:
294:
290:
286:
282:
280:
276:
275:
274:
270:
266:
262:
261:
260:
256:
255:
254:
253:
228:
224:
215:
212:
205:
204:Terminal ring
201:
178:
174:
173:
172:
168:
166:
162:
160:
156:
154:
150:
148:
144:
143:
142:
141:
140:
133:
129:
127:
123:
121:
117:
116:
115:
114:
113:
106:
102:
100:
96:
94:
90:
86:
82:
80:
76:
75:
74:
73:Quotient ring
70:
68:
64:
62:
58:
57:
56:
55:
46:
45:
42:
37:β Ring theory
36:
32:
31:
19:
2759:
2736:
2709:
2671:
2659:
2656:Bourbaki, N.
2640:
2624:
2599:
2594:, p. 88
2587:
2575:
2570:, p. 12
2563:
2551:
2531:
2499:
2495:
2491:epimorphisms
2488:
2482:
2473:
2469:
2460:
2456:
2448:
2441:
2437:
2433:
2429:
2422:
2415:
2411:
2404:
2397:
2392:
2388:
2384:
2377:
2366:
2355:
2351:
2347:
2343:
2339:
2331:
2324:
2298:
2294:
2289:
2285:
2281:
2277:
2272:
2268:
2264:
2259:
2255:
2246:
2240:
2236:
2233:
2229:
2221:
2217:
2213:
2209:
2205:
2201:
2188:
2182:
2178:
2174:
2170:
2166:
2159:Non-examples
2149:
2144:
2127:
2123:
2118:
2117:over a ring
2114:
2110:
2105:
2101:
2097:
2093:
2086:
2082:
2078:
2073:
2069:
2060:
2055:
2050:
2045:
2040:matrix rings
2035:
2031:
2027:
2022:
2018:
2014:
2003:
1997:
1993:
1989:
1985:
1981:
1975:
1971:
1967:
1963:
1957:
1953:
1949:
1943:denotes the
1940:
1936:
1934:real numbers
1929:
1921:
1913:
1907:
1898:
1894:
1890:
1886:
1882:
1869:
1864:
1860:
1856:
1851:
1847:
1839:
1835:
1820:
1817:
1808:
1804:
1799:
1794:
1790:
1784:
1781:
1777:
1773:
1769:
1743:
1739:
1723:
1719:
1714:
1703:
1699:
1694:
1690:
1674:
1670:
1665:
1657:
1653:
1647:
1643:
1637:
1633:
1626:
1620:
1616:
1612:
1607:
1603:
1597:
1593:
1588:
1580:
1572:
1567:
1562:
1558:
1554:
1547:
1543:
1535:
1531:
1527:
1520:
1516:
1511:
1505:
1500:
1492:
1488:
1484:
1477:
1473:
1468:
1463:
1455:
1451:
1447:
1440:
1436:
1431:
1426:
1422:
1415:
1407:
1403:
1399:
1391:
1387:
1379:
1370:
1358:
1347:
1340:
1333:
1327:
1323:
1319:
1314:
1307:
1303:
1292:
1284:
1280:
1275:
1271:
1267:
1260:
1242:
1238:
1227:
1222:
1218:
1214:
1210:
1206:
1200:
1189:
1185:
1181:
1170:
1166:
1162:
1154:
1148:
1144:
1140:
1136:
1131:
1127:
1123:
1119:
1109:
1105:
1101:
1097:
1093:
1089:
1082:
1076:
1071:
1063:
1059:
1055:
1052:
1028:
1022:
1016:
1011:
1007:
999:
995:
993:
988:
984:
980:
976:
975:is called a
971:
965:
952:
949:
946:
895:
675:
671:
666:between two
659:
653:
613:
599:
598:
594:Free algebra
592:
591:
585:
584:
553:
496:
458:
421:
390:
389:
369:Finite field
318:
265:Finite field
251:
250:
177:Initial ring
137:
136:
111:
110:
109:
52:
2799:Ring theory
2756:Lang, Serge
2682:. xvi+785.
2662:. Springer.
2606:, p. 3
2362:of order 4.
2354:are called
2208:and image 2
2186:defined by
1961:defined by
1926:polynomials
1846:For a ring
1755:zero object
1542:, then ker(
1460:prime ideal
1373:is not the
1031:composition
987:are called
960:, then its
714:; that is,
656:mathematics
574:Simple ring
285:Jordan ring
159:Graded ring
41:Ring theory
2793:Categories
2613:References
2556:Artin 1991
2356:isomorphic
1814:surjective
1627:Moreover,
1398:, then im(
1367:skew-field
1250:zero ideal
1169:(or of im(
1157:induces a
1104:) for all
1049:Properties
989:isomorphic
580:Commutator
339:GCD domain
2804:Morphisms
2592:Lang 2002
2544:Citations
2436:that map
2336:bijective
2135:A unital
2091:given by
1903:zero ring
1735:of rings.
1708:zero ring
1375:zero ring
1039:morphisms
958:bijection
695:→
521:∞
299:Semifield
2758:(2002),
2734:(1985).
2708:(2004).
2670:(1995).
2658:(1998).
2639:(1969),
2623:(1991).
2511:See also
2387: :
2244:for any
2169: :
2081: :
2017: :
1952: :
1772: :
1762:Examples
1733:category
1386:If both
1339: :
1322: :
1115:For any
1058: :
1035:category
896:for all
664:function
293:Semiring
279:Lie ring
61:Subrings
2782:1878556
2760:Algebra
2698:1322960
2649:0242802
2625:Algebra
2346:, then
2152:-linear
2034:, then
1901:is the
1731:in the
1706:is the
1611:, then
1575:) is a
1377:, then
1300:subring
1288:exists.
1264:divides
1233:, is a
1217:|
962:inverse
495:PrΓΌfer
97:β’
2780:
2770:
2744:
2720:
2696:
2686:
2647:
2126:β End(
2085:β End(
1939:, and
1538:is an
1503:, and
1396:fields
1369:) and
1197:kernel
147:Module
120:Kernel
2523:Notes
2428:from
2053:) β M
1823:(see
1812:is a
1495:is a
1466:then
1458:is a
1429:then
1406:, so
1363:field
1361:is a
1225:) = 0
1209:) = {
1178:image
1096:) = β
1080:) = 0
1041:(see
1020:) = 1
956:is a
668:rings
499:-ring
363:Field
259:Field
67:Ideal
54:Rings
2768:ISBN
2742:ISBN
2718:ISBN
2684:ISBN
2467:and
2447:and
2421:and
2403:and
2360:rngs
2350:and
2342:and
2280:to (
2258:and
2195:) =
2068:Let
2030:and
1970:) =
1911:to 1
1885:and
1830:The
1807:mod
1797:) =
1641:and
1606:) β
1602:ker(
1600:and
1557:and
1530:and
1487:and
1450:and
1394:are
1390:and
1306:and
1274:and
1255:The
1205:ker(
1195:The
1176:The
1143:) =
1117:unit
1053:Let
1029:The
1004:rngs
1002:are
998:and
983:and
674:and
658:, a
2440:to
2432:to
2254:If
2249:β₯ 1
2012:If
2007:+ 1
1920:If
1893:to
1881:If
1586:If
1579:of
1553:If
1526:If
1499:of
1483:If
1462:of
1446:If
1421:If
1414:of
1357:If
1291:If
1259:of
1237:in
1213:in
1199:of
1180:of
1173:)).
1122:in
1108:in
994:If
922:in
654:In
2795::
2778:MR
2776:,
2762:,
2716:.
2712:.
2694:MR
2692:.
2674:.
2645:MR
2635:;
2507:.
2498:β
2472:β
2459:β
2391:β
2365:A
2330:A
2323:A
2297:Γ
2271:Γ
2267:β
2239:β
2224:).
2220:/3
2212:/6
2204:/6
2181:/6
2177:β
2173:/6
2106:av
2104:=
2021:β
1956:β
1867:β
1863:,
1859:β
1854:,
1838:β
1827:).
1803:=
1776:β
1722:β
1693:β
1673:β
1656:β
1646:β
1636:β
1346:β
1326:β
1283:β
1126:,
1092:(β
1074:(0
1062:β
1014:(1
607:β’
578:β’
572:β’
566:β’
560:β’
493:β’
456:β’
419:β’
413:β’
404:β’
398:β’
381:β’
375:β’
367:β’
361:β’
355:β’
349:β’
343:β’
337:β’
331:β’
325:β’
297:β’
291:β’
283:β’
277:β’
271:β’
263:β’
257:β’
202:β’
175:β’
169:β’
163:β’
157:β’
151:β’
145:β’
130:β’
124:β’
118:β’
103:β’
91:β’
83:β’
77:β’
71:β’
65:β’
59:β’
2750:.
2726:.
2700:.
2500:Q
2496:Z
2483:f
2477:2
2474:g
2470:f
2464:1
2461:g
2457:f
2452:2
2449:r
2445:1
2442:r
2438:x
2434:R
2430:Z
2426:2
2423:g
2419:1
2416:g
2412:S
2408:2
2405:r
2401:1
2398:r
2393:S
2389:R
2385:f
2352:S
2348:R
2344:S
2340:R
2302:.
2299:S
2295:R
2290:R
2286:S
2282:r
2278:r
2273:S
2269:R
2265:R
2260:S
2256:R
2251:.
2247:n
2241:Z
2237:Z
2234:n
2232:/
2230:Z
2222:Z
2218:Z
2214:Z
2210:Z
2206:Z
2202:Z
2197:6
2193:6
2191:(
2189:f
2183:Z
2179:Z
2175:Z
2171:Z
2167:f
2154:.
2150:R
2145:R
2132:.
2130:)
2128:M
2124:R
2119:R
2115:M
2111:M
2102:v
2100:)
2098:a
2096:(
2094:Ο
2089:)
2087:V
2083:k
2079:Ο
2074:k
2070:V
2065:.
2063:)
2061:S
2059:(
2056:n
2051:R
2049:(
2046:n
2043:M
2036:f
2032:S
2028:R
2023:S
2019:R
2015:f
2009:.
2004:X
1998:R
1994:f
1990:p
1986:X
1982:i
1978:)
1976:i
1974:(
1972:p
1968:p
1966:(
1964:f
1958:C
1954:R
1950:f
1941:C
1937:R
1930:X
1922:R
1914:S
1908:R
1899:S
1895:S
1891:R
1887:S
1883:R
1878:.
1870:x
1865:x
1861:R
1857:R
1852:p
1848:R
1840:C
1836:C
1821:Z
1818:n
1809:n
1805:a
1800:n
1795:a
1793:(
1791:f
1785:Z
1782:n
1780:/
1778:Z
1774:Z
1770:f
1744:R
1740:R
1724:R
1720:Z
1715:R
1704:S
1700:R
1695:S
1691:R
1685:.
1675:R
1671:R
1666:R
1661:.
1658:T
1654:R
1648:S
1644:R
1638:T
1634:S
1623:.
1621:S
1617:P
1615:(
1613:f
1608:P
1604:f
1598:R
1594:P
1589:f
1583:.
1581:R
1573:f
1568:f
1563:S
1559:S
1555:R
1550:.
1548:R
1544:f
1536:S
1532:S
1528:R
1523:.
1521:R
1517:M
1515:(
1512:f
1506:f
1501:S
1493:M
1489:S
1485:R
1480:.
1478:R
1474:P
1472:(
1469:f
1464:S
1456:P
1452:S
1448:R
1443:.
1441:R
1437:I
1435:(
1432:f
1427:S
1423:I
1418:.
1416:R
1408:S
1404:S
1400:f
1392:S
1388:R
1380:f
1371:S
1359:R
1354:.
1350:p
1348:S
1343:p
1341:R
1336:p
1334:f
1328:S
1324:R
1320:f
1315:S
1310:p
1308:S
1304:R
1295:p
1293:R
1285:S
1281:R
1276:S
1272:R
1268:R
1261:S
1252:.
1243:R
1239:R
1231:}
1228:S
1223:a
1221:(
1219:f
1215:R
1211:a
1207:f
1201:f
1192:.
1190:S
1186:f
1182:f
1171:f
1167:S
1163:R
1155:f
1151:)
1149:a
1147:(
1145:f
1141:a
1139:(
1137:f
1132:a
1130:(
1128:f
1124:R
1120:a
1112:.
1110:R
1106:a
1102:a
1100:(
1098:f
1094:a
1090:f
1086:.
1083:S
1077:R
1072:f
1064:S
1060:R
1056:f
1023:S
1017:R
1012:f
1000:S
996:R
985:S
981:R
972:f
966:f
953:f
933:.
930:R
910:b
907:,
904:a
877:,
872:S
868:1
864:=
857:)
852:R
848:1
844:(
841:f
834:,
831:)
828:b
825:(
822:f
819:)
816:a
813:(
810:f
807:=
800:)
797:b
794:a
791:(
788:f
781:,
778:)
775:b
772:(
769:f
766:+
763:)
760:a
757:(
754:f
751:=
744:)
741:b
738:+
735:a
732:(
729:f
698:S
692:R
689::
686:f
676:S
672:R
643:e
636:t
629:v
526:)
517:p
513:(
509:Z
497:p
477:p
472:Q
459:p
440:p
435:Z
422:p
408:n
233:Z
229:1
225:/
220:Z
216:=
213:0
187:Z
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.