2347:. 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
2524:
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,
880:
714:
233:
525:
709:
478:
441:
187:
697:
909:
2323:
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
932:
630:
2760:
87:
2734:
2710:
2676:
623:
575:
197:
875:{\displaystyle {\begin{aligned}f(a+b)&=f(a)+f(b),\\f(ab)&=f(a)f(b),\\f(1_{R})&=1_{S},\end{aligned}}}
492:
2752:
2664:
616:
483:
1034:). In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.
333:
93:
454:
417:
108:
1746:
in the category of rings; in particular, the zero ring is not a zero object in the category of rings.
1245:
568:
371:
321:
170:
700:
380:
114:
73:
1864:
1721:
1023:
652:
537:
388:
339:
120:
670:
2327:
as a function on the underlying sets. If there exists a ring isomorphism between two rings
2787:
2770:
2686:
2637:
2125:
1019:
936:
These conditions imply that additive inverses and the additive identity are preserved too.
261:
135:
8:
2137:
2129:
1820:
1670:
Therefore, the class of all rings together with ring homomorphisms forms a category, the
1384:
1351:
1166:
980:. From the standpoint of ring theory, isomorphic rings have exactly the same properties.
888:
543:
351:
302:
247:
141:
127:
55:
23:
914:
2792:
2694:
2493:
2492:
is a ring epimorphism, but not a surjection. However, they are exactly the same as the
1813:
1147:
1105:
656:
556:
42:
2724:
2525:
explicitly specify that rings are unital and that homomorphisms preserve the identity.
2756:
2730:
2706:
2672:
2301:
1671:
1031:
597:
394:
159:
1022:
of two ring homomorphisms is a ring homomorphism. It follows that the rings forms a
2644:
2625:
1185:
950:
603:
589:
403:
345:
308:
81:
67:
2766:
2720:
2702:
2682:
2668:
2633:
2505:
1736:
1528:
1400:
1223:
365:
315:
153:
2656:
2621:
1933:
1717:
1056:
be a ring homomorphism. Then, directly from these definitions, one can deduce:
409:
2277:
is not the zero ring), since it does not map the multiplicative identity 1 of
1015:. A rng homomorphism between (unital) rings need not be a ring homomorphism.
2781:
2609:
2348:
1565:
1485:
992:
550:
446:
61:
1742:
As the initial object is not isomorphic to the terminal object, there is no
2368:
582:
357:
253:
2479:
2028:
1922:
1743:
1448:
644:
562:
273:
147:
29:
2632:, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.,
2744:
1914:
1802:
1355:
1238:
327:
1906:). On the other hand, the zero function is always a rng homomorphism.
2755:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
2324:
1891:
1696:
1363:
946:
287:
192:
1832:
is a ring homomorphism (this is an example of a ring automorphism).
1027:
281:
267:
2478:
However, surjective ring homomorphisms are vastly different from
2306:
1288:
1259:. This can sometimes be used to show that between certain rings
1252:
165:
49:
2098:
is a ring homomorphism. More generally, given an abelian group
2385:
is a monomorphism that is not injective, then it sends some
2273:,0) is a rng homomorphism, but not a ring homomorphism (if
2189:
is a rng homomorphism (and rng endomorphism), with kernel 3
1237:
An homomorphism is injective if and only if kernel is the
2661:
Commutative algebra with a view toward algebraic geometry
2586:
2562:
2482:
in the category of rings. For example, the inclusion
999:, defined as above except without the third condition
2550:
1735:
to the zero ring. This says that the zero ring is a
917:
891:
712:
673:
495:
457:
420:
200:
173:
2538:
1981:) is a surjective ring homomorphism. The kernel of
926:
903:
874:
691:
667:are rings, then a ring homomorphism is a function
519:
472:
435:
227:
181:
2574:
2362:
2779:
2620:
16:Structure-preserving function between two rings
2367:Injective ring homomorphisms are identical to
2307:Endomorphisms, isomorphisms, and automorphisms
2316:is a ring homomorphism from a ring to itself.
995:, then the corresponding notion is that of a
624:
2358:is a ring isomorphism from a ring to itself.
2110:is equivalent to giving a ring homomorphism
1716:. This says that the ring of integers is an
1150:from the (multiplicative) group of units of
958:is also a ring homomorphism. In this case,
699:that preserves addition, multiplication and
228:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
1731:, there is a unique ring homomorphism from
2693:
2592:
1154:to the (multiplicative) group of units of
631:
617:
2015:is a ring homomorphism between the rings
520:{\displaystyle \mathbb {Z} (p^{\infty })}
497:
460:
423:
221:
208:
175:
2719:
2655:
2643:
2616:. Englewood Cliffs, N.J.: Prentice Hall.
2568:
2556:
2281:to the multiplicative identity (1,1) of
2027:induces a ring homomorphism between the
1234:is the kernel of some ring homomorphism.
2780:
2475:is a monomorphism this is impossible.
1886:is a ring homomorphism if and only if
1706:, there is a unique ring homomorphism
1699:(the ring whose only element is zero).
1620:The composition of ring homomorphisms
2608:
2544:
1302:is the smallest subring contained in
2743:
2580:
2295:
2136:is a ring homomorphism that is also
1691:to 0 is a ring homomorphism only if
88:Free product of associative algebras
2630:Introduction to commutative algebra
13:
1878:are rings, the zero function from
1863:is a ring endomorphism called the
1230:. Every two-sided ideal in a ring
509:
14:
2804:
576:Noncommutative algebraic geometry
473:{\displaystyle \mathbb {Q} _{p}}
436:{\displaystyle \mathbb {Z} _{p}}
2147:
2061:be a vector space over a field
1985:consists of all polynomials in
1969:(substitute the imaginary unit
1306:, then every ring homomorphism
2518:
2363:Monomorphisms and epimorphisms
2216:There is no ring homomorphism
1805:ring homomorphism with kernel
1608:) is prime (maximal) ideal in
1123:) is a unit element such that
845:
832:
819:
813:
807:
801:
788:
779:
766:
760:
751:
745:
732:
720:
683:
514:
501:
1:
2753:Graduate Texts in Mathematics
2665:Graduate Texts in Mathematics
2601:
2371:in the category of rings: If
1037:
2532:
1894:(otherwise it fails to map 1
1687:that sends every element of
1585:is prime (maximal) ideal in
1320:induces a ring homomorphism
182:{\displaystyle \mathbb {Z} }
7:
2699:Algebras, rings and modules
2667:. Vol. 150. New York:
2499:
1750:
1026:with ring homomorphisms as
334:Unique factorization domain
10:
2809:
2299:
651:is a structure-preserving
94:Tensor product of algebras
2469:are identical, but since
2403:. Consider the two maps
2251:are rings, the inclusion
1921:with coefficients in the
1739:in the category of rings.
2511:
2205:(which is isomorphic to
2132:over a commutative ring
2102:, a module structure on
1913:denotes the ring of all
1839:of prime characteristic
1560:is surjective, then ker(
1508:) is a maximal ideal of
692:{\displaystyle f:R\to S}
372:Formal power series ring
322:Integrally closed domain
2649:Algebra I, Chapters 1β3
2399:to the same element of
1667:is a ring homomorphism.
1640:is a ring homomorphism
1267:, no ring homomorphism
701:multiplicative identity
381:Algebraic number theory
74:Total ring of fractions
1989:that are divisible by
1865:Frobenius endomorphism
1535:) is a prime ideal of
1465:) is a prime ideal of
1255:the characteristic of
928:
905:
876:
693:
659:. More explicitly, if
538:Noncommutative algebra
521:
474:
437:
389:Algebraic number field
340:Principal ideal domain
229:
183:
121:Frobenius endomorphism
1354:(or more generally a
929:
906:
877:
694:
522:
475:
438:
230:
184:
2130:associative algebras
2126:algebra homomorphism
1936:, then the function
1523:are commutative and
1498:is surjective, then
1443:are commutative and
915:
889:
710:
671:
544:Noncommutative rings
493:
455:
418:
262:Non-associative ring
198:
171:
128:Algebraic structures
2695:Hazewinkel, Michiel
2494:strong epimorphisms
1821:complex conjugation
1657:, the identity map
1399:can be viewed as a
1391:) is a subfield of
1177:), is a subring of
904:{\displaystyle a,b}
303:Commutative algebra
142:Associative algebra
24:Algebraic structure
2622:Atiyah, Michael F.
1977:in the polynomial
1814:modular arithmetic
1148:group homomorphism
927:{\displaystyle R.}
924:
901:
872:
870:
689:
557:Semiprimitive ring
517:
470:
433:
241:Related structures
225:
179:
115:Inner automorphism
101:Ring homomorphisms
2762:978-0-387-95385-4
2626:Macdonald, Ian G.
2356:ring automorphism
2314:ring endomorphism
2302:Category of rings
2296:Category of rings
1973:for the variable
1672:category of rings
1550:are commutative,
1480:are commutative,
1428:) is an ideal of
1142:. In particular,
1032:Category of rings
649:ring homomorphism
641:
640:
598:Geometric algebra
309:Commutative rings
160:Category of rings
2800:
2773:
2740:
2729:(2nd ed.).
2721:Jacobson, Nathan
2716:
2690:
2652:
2640:
2617:
2596:
2590:
2584:
2578:
2572:
2566:
2560:
2554:
2548:
2542:
2526:
2522:
2491:
2474:
2468:
2455:
2443:, respectively;
2384:
2321:ring isomorphism
2290:
2265:that sends each
2264:
2239:
2232:
2188:
2174:
2120:
2097:
2079:
2053:
2014:
1997:
1995:
1968:
1949:
1917:in the variable
1862:
1861:
1831:
1800:
1776:
1715:
1686:
1666:
1649:
1639:
1629:
1599:
1580:
1559:
1554:is a field, and
1503:
1497:
1460:
1423:
1371:
1342:
1319:
1287:is the smallest
1276:
1221:
1141:
1055:
997:rng homomorphism
968:, and the rings
966:ring isomorphism
963:
957:
944:
933:
931:
930:
925:
910:
908:
907:
902:
881:
879:
878:
873:
871:
864:
863:
844:
843:
698:
696:
695:
690:
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:
2808:
2807:
2803:
2802:
2801:
2799:
2798:
2797:
2778:
2777:
2776:
2763:
2737:
2726:Basic algebra I
2713:
2703:Springer-Verlag
2679:
2669:Springer-Verlag
2657:Eisenbud, David
2604:
2599:
2593:Hazewinkel 2004
2591:
2587:
2579:
2575:
2567:
2563:
2555:
2551:
2543:
2539:
2535:
2530:
2529:
2523:
2519:
2514:
2506:Change of rings
2502:
2483:
2470:
2467:
2457:
2454:
2444:
2442:
2435:
2416:
2409:
2398:
2391:
2372:
2365:
2309:
2304:
2298:
2282:
2252:
2234:
2217:
2187:
2183:
2176:
2154:
2150:
2128:between unital
2111:
2081:
2066:
2065:. Then the map
2047:
2037:
2031:
2002:
1991:
1990:
1951:
1937:
1934:complex numbers
1905:
1899:
1857:
1844:
1823:
1791:
1778:
1757:
1753:
1737:terminal object
1727:For every ring
1707:
1702:For every ring
1678:
1658:
1641:
1631:
1621:
1590:
1581:is surjective,
1576:
1555:
1529:integral domain
1499:
1493:
1456:
1419:
1414:is an ideal of
1401:field extension
1367:
1340:
1333:
1326:
1321:
1307:
1300:
1285:
1268:
1224:two-sided ideal
1219:
1193:
1124:
1074:
1068:
1043:
1040:
1014:
1008:
959:
953:
940:
939:If in addition
916:
913:
912:
890:
887:
886:
869:
868:
859:
855:
848:
839:
835:
826:
825:
791:
773:
772:
735:
713:
711:
708:
707:
672:
669:
668:
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:
2806:
2796:
2795:
2790:
2775:
2774:
2761:
2741:
2735:
2717:
2711:
2691:
2677:
2653:
2641:
2618:
2610:Artin, Michael
2605:
2603:
2600:
2598:
2597:
2585:
2573:
2561:
2549:
2536:
2534:
2531:
2528:
2527:
2516:
2515:
2513:
2510:
2509:
2508:
2501:
2498:
2465:
2452:
2440:
2433:
2414:
2407:
2396:
2389:
2364:
2361:
2360:
2359:
2352:
2317:
2308:
2305:
2300:Main article:
2297:
2294:
2293:
2292:
2241:
2214:
2185:
2181:
2149:
2146:
2145:
2144:
2122:
2055:
2043:
2033:
1999:
1907:
1901:
1895:
1868:
1833:
1817:
1787:
1752:
1749:
1748:
1747:
1740:
1725:
1718:initial object
1700:
1675:
1668:
1653:For each ring
1651:
1614:
1613:
1573:
1540:
1513:
1470:
1433:
1408:
1373:
1344:
1338:
1331:
1324:
1298:
1283:
1278:
1246:characteristic
1242:
1235:
1215:
1182:
1163:
1102:
1076:
1070:
1064:
1039:
1036:
1010:
1004:
923:
920:
900:
897:
894:
883:
882:
867:
862:
858:
854:
851:
849:
847:
842:
838:
834:
831:
828:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
792:
790:
787:
784:
781:
778:
775:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
736:
734:
731:
728:
725:
722:
719:
716:
715:
688:
685:
682:
679:
676:
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:
2805:
2794:
2791:
2789:
2786:
2785:
2783:
2772:
2768:
2764:
2758:
2754:
2750:
2746:
2742:
2738:
2736:9780486471891
2732:
2728:
2727:
2722:
2718:
2714:
2712:1-4020-2690-0
2708:
2704:
2700:
2696:
2692:
2688:
2684:
2680:
2678:0-387-94268-8
2674:
2670:
2666:
2662:
2658:
2654:
2650:
2646:
2642:
2639:
2635:
2631:
2627:
2623:
2619:
2615:
2611:
2607:
2606:
2594:
2589:
2582:
2577:
2571:, p. 103
2570:
2569:Jacobson 1985
2565:
2558:
2557:Eisenbud 1995
2553:
2547:, p. 353
2546:
2541:
2537:
2521:
2517:
2507:
2504:
2503:
2497:
2495:
2490:
2486:
2481:
2476:
2473:
2464:
2460:
2451:
2447:
2439:
2432:
2428:
2424:
2420:
2413:
2406:
2402:
2395:
2388:
2383:
2379:
2375:
2370:
2369:monomorphisms
2357:
2353:
2350:
2346:
2342:
2338:
2334:
2330:
2326:
2322:
2318:
2315:
2311:
2310:
2303:
2289:
2285:
2280:
2276:
2272:
2268:
2263:
2259:
2255:
2250:
2246:
2242:
2237:
2231:
2227:
2224:
2220:
2215:
2212:
2208:
2204:
2200:
2196:
2192:
2179:
2173:
2169:
2165:
2161:
2157:
2153:The function
2152:
2151:
2142:
2140:
2135:
2131:
2127:
2123:
2118:
2114:
2109:
2105:
2101:
2096:
2092:
2088:
2084:
2077:
2073:
2069:
2064:
2060:
2056:
2051:
2046:
2041:
2036:
2030:
2026:
2022:
2018:
2013:
2009:
2005:
2000:
1994:
1988:
1984:
1980:
1976:
1972:
1966:
1962:
1958:
1954:
1948:
1944:
1940:
1935:
1931:
1927:
1924:
1920:
1916:
1912:
1908:
1904:
1898:
1893:
1889:
1885:
1881:
1877:
1873:
1869:
1866:
1860:
1855:
1851:
1847:
1842:
1838:
1834:
1830:
1826:
1822:
1818:
1815:
1811:
1808:
1804:
1799:
1795:
1790:
1785:
1781:
1777:, defined by
1775:
1772:
1768:
1764:
1760:
1756:The function
1755:
1754:
1745:
1741:
1738:
1734:
1730:
1726:
1723:
1719:
1714:
1710:
1705:
1701:
1698:
1694:
1690:
1685:
1681:
1677:The zero map
1676:
1673:
1669:
1665:
1661:
1656:
1652:
1648:
1644:
1638:
1634:
1628:
1624:
1619:
1618:
1617:
1611:
1607:
1603:
1598:
1594:
1588:
1584:
1579:
1574:
1571:
1567:
1566:maximal ideal
1563:
1558:
1553:
1549:
1545:
1541:
1538:
1534:
1530:
1526:
1522:
1518:
1514:
1511:
1507:
1502:
1496:
1491:
1487:
1486:maximal ideal
1483:
1479:
1475:
1471:
1468:
1464:
1459:
1454:
1450:
1446:
1442:
1438:
1434:
1431:
1427:
1422:
1417:
1413:
1409:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1374:
1372:is injective.
1370:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1334:
1327:
1318:
1314:
1310:
1305:
1301:
1294:
1291:contained in
1290:
1286:
1279:
1275:
1271:
1266:
1262:
1258:
1254:
1251:
1247:
1243:
1240:
1236:
1233:
1229:
1225:
1218:
1213:
1209:
1205:
1201:
1197:
1192:, defined as
1191:
1187:
1183:
1180:
1176:
1173:, denoted im(
1172:
1168:
1164:
1161:
1157:
1153:
1149:
1145:
1139:
1135:
1131:
1127:
1122:
1118:
1114:
1110:
1107:
1103:
1100:
1096:
1092:
1088:
1084:
1080:
1077:
1073:
1067:
1062:
1059:
1058:
1057:
1054:
1050:
1046:
1035:
1033:
1029:
1025:
1021:
1016:
1013:
1007:
1002:
998:
994:
990:
986:
981:
979:
975:
971:
967:
962:
956:
952:
948:
943:
937:
934:
921:
918:
898:
895:
892:
865:
860:
856:
852:
850:
840:
836:
829:
822:
816:
810:
804:
798:
795:
793:
785:
782:
776:
769:
763:
757:
754:
748:
742:
739:
737:
729:
726:
723:
717:
706:
705:
704:
702:
686:
680:
677:
674:
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:β Ring theory
25:
21:
20:
2748:
2725:
2698:
2660:
2648:
2645:Bourbaki, N.
2629:
2613:
2588:
2583:, p. 88
2576:
2564:
2559:, p. 12
2552:
2540:
2520:
2488:
2484:
2480:epimorphisms
2477:
2471:
2462:
2458:
2449:
2445:
2437:
2430:
2426:
2422:
2418:
2411:
2404:
2400:
2393:
2386:
2381:
2377:
2373:
2366:
2355:
2344:
2340:
2336:
2332:
2328:
2320:
2313:
2287:
2283:
2278:
2274:
2270:
2266:
2261:
2257:
2253:
2248:
2244:
2235:
2229:
2225:
2222:
2218:
2210:
2206:
2202:
2198:
2194:
2190:
2177:
2171:
2167:
2163:
2159:
2155:
2148:Non-examples
2138:
2133:
2116:
2112:
2107:
2106:over a ring
2103:
2099:
2094:
2090:
2086:
2082:
2075:
2071:
2067:
2062:
2058:
2049:
2044:
2039:
2034:
2029:matrix rings
2024:
2020:
2016:
2011:
2007:
2003:
1992:
1986:
1982:
1978:
1974:
1970:
1964:
1960:
1956:
1952:
1946:
1942:
1938:
1932:denotes the
1929:
1925:
1923:real numbers
1918:
1910:
1902:
1896:
1887:
1883:
1879:
1875:
1871:
1858:
1853:
1849:
1845:
1840:
1836:
1828:
1824:
1809:
1806:
1797:
1793:
1788:
1783:
1779:
1773:
1770:
1766:
1762:
1758:
1732:
1728:
1712:
1708:
1703:
1692:
1688:
1683:
1679:
1663:
1659:
1654:
1646:
1642:
1636:
1632:
1626:
1622:
1615:
1609:
1605:
1601:
1596:
1592:
1586:
1582:
1577:
1569:
1561:
1556:
1551:
1547:
1543:
1536:
1532:
1524:
1520:
1516:
1509:
1505:
1500:
1494:
1489:
1481:
1477:
1473:
1466:
1462:
1457:
1452:
1444:
1440:
1436:
1429:
1425:
1420:
1415:
1411:
1404:
1396:
1392:
1388:
1380:
1376:
1368:
1359:
1347:
1336:
1329:
1322:
1316:
1312:
1308:
1303:
1296:
1292:
1281:
1273:
1269:
1264:
1260:
1256:
1249:
1231:
1227:
1216:
1211:
1207:
1203:
1199:
1195:
1189:
1178:
1174:
1170:
1159:
1155:
1151:
1143:
1137:
1133:
1129:
1125:
1120:
1116:
1112:
1108:
1098:
1094:
1090:
1086:
1082:
1078:
1071:
1065:
1060:
1052:
1048:
1044:
1041:
1017:
1011:
1005:
1000:
996:
988:
984:
982:
977:
973:
969:
965:
964:is called a
960:
954:
941:
938:
935:
884:
664:
660:
655:between two
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:
100:
99:
98:
41:
2788:Ring theory
2745:Lang, Serge
2671:. xvi+785.
2651:. Springer.
2595:, p. 3
2351:of order 4.
2343:are called
2197:and image 2
2175:defined by
1950:defined by
1915:polynomials
1835:For a ring
1744:zero object
1531:, then ker(
1449:prime ideal
1362:is not the
1020:composition
976:are called
949:, then its
703:; that is,
645:mathematics
563:Simple ring
274:Jordan ring
148:Graded ring
30:Ring theory
2782:Categories
2602:References
2545:Artin 1991
2345:isomorphic
1803:surjective
1616:Moreover,
1387:, then im(
1356:skew-field
1239:zero ideal
1158:(or of im(
1146:induces a
1093:) for all
1038:Properties
978:isomorphic
569:Commutator
328:GCD domain
2793:Morphisms
2581:Lang 2002
2533:Citations
2425:that map
2325:bijective
2124:A unital
2080:given by
1892:zero ring
1724:of rings.
1697:zero ring
1364:zero ring
1028:morphisms
947:bijection
684:→
510:∞
288:Semifield
2747:(2002),
2723:(1985).
2697:(2004).
2659:(1995).
2647:(1998).
2628:(1969),
2612:(1991).
2500:See also
2376: :
2233:for any
2158: :
2070: :
2006: :
1941: :
1761: :
1751:Examples
1722:category
1375:If both
1328: :
1311: :
1104:For any
1047: :
1024:category
885:for all
653:function
282:Semiring
268:Lie ring
50:Subrings
2771:1878556
2749:Algebra
2687:1322960
2638:0242802
2614:Algebra
2335:, then
2141:-linear
2023:, then
1890:is the
1720:in the
1695:is the
1600:, then
1564:) is a
1366:, then
1289:subring
1277:exists.
1253:divides
1222:, is a
1206:|
951:inverse
484:PrΓΌfer
86:β’
2769:
2759:
2733:
2709:
2685:
2675:
2636:
2115:β End(
2074:β End(
1928:, and
1527:is an
1492:, and
1385:fields
1358:) and
1186:kernel
136:Module
109:Kernel
2512:Notes
2417:from
2042:) β M
1812:(see
1801:is a
1484:is a
1455:then
1447:is a
1418:then
1395:, so
1352:field
1350:is a
1214:) = 0
1198:) = {
1167:image
1085:) = β
1069:) = 0
1030:(see
1009:) = 1
945:is a
657:rings
488:-ring
352:Field
248:Field
56:Ideal
43:Rings
2757:ISBN
2731:ISBN
2707:ISBN
2673:ISBN
2456:and
2436:and
2410:and
2392:and
2349:rngs
2339:and
2331:and
2269:to (
2247:and
2184:) =
2057:Let
2019:and
1959:) =
1900:to 1
1874:and
1819:The
1796:mod
1786:) =
1630:and
1595:) β
1591:ker(
1589:and
1546:and
1519:and
1476:and
1439:and
1383:are
1379:and
1295:and
1263:and
1244:The
1194:ker(
1184:The
1165:The
1132:) =
1106:unit
1042:Let
1018:The
993:rngs
991:are
987:and
972:and
663:and
647:, a
2429:to
2421:to
2243:If
2238:β₯ 1
2001:If
1996:+ 1
1909:If
1882:to
1870:If
1575:If
1568:of
1542:If
1515:If
1488:of
1472:If
1451:of
1435:If
1410:If
1403:of
1346:If
1280:If
1248:of
1226:in
1202:in
1188:of
1169:of
1162:)).
1111:in
1097:in
983:If
911:in
643:In
2784::
2767:MR
2765:,
2751:,
2705:.
2701:.
2683:MR
2681:.
2663:.
2634:MR
2624:;
2496:.
2487:β
2461:β
2448:β
2380:β
2354:A
2319:A
2312:A
2286:Γ
2260:Γ
2256:β
2228:β
2213:).
2209:/3
2201:/6
2193:/6
2170:/6
2166:β
2162:/6
2095:av
2093:=
2010:β
1945:β
1856:β
1852:,
1848:β
1843:,
1827:β
1816:).
1792:=
1765:β
1711:β
1682:β
1662:β
1645:β
1635:β
1625:β
1335:β
1315:β
1272:β
1115:,
1081:(β
1063:(0
1051:β
1003:(1
596:β’
567:β’
561:β’
555:β’
549:β’
482:β’
445:β’
408:β’
402:β’
393:β’
387:β’
370:β’
364:β’
356:β’
350:β’
344:β’
338:β’
332:β’
326:β’
320:β’
314:β’
286:β’
280:β’
272:β’
266:β’
260:β’
252:β’
246:β’
191:β’
164:β’
158:β’
152:β’
146:β’
140:β’
134:β’
119:β’
113:β’
107:β’
92:β’
80:β’
72:β’
66:β’
60:β’
54:β’
48:β’
2739:.
2715:.
2689:.
2489:Q
2485:Z
2472:f
2466:2
2463:g
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2446:f
2441:2
2438:r
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2431:r
2427:x
2423:R
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2412:g
2408:1
2405:g
2401:S
2397:2
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2390:1
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2378:R
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2341:S
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2333:S
2329:R
2291:.
2288:S
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2279:R
2275:S
2271:r
2267:r
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2254:R
2249:S
2245:R
2240:.
2236:n
2230:Z
2226:Z
2223:n
2221:/
2219:Z
2211:Z
2207:Z
2203:Z
2199:Z
2195:Z
2191:Z
2186:6
2182:6
2180:(
2178:f
2172:Z
2168:Z
2164:Z
2160:Z
2156:f
2143:.
2139:R
2134:R
2121:.
2119:)
2117:M
2113:R
2108:R
2104:M
2100:M
2091:v
2089:)
2087:a
2085:(
2083:Ο
2078:)
2076:V
2072:k
2068:Ο
2063:k
2059:V
2054:.
2052:)
2050:S
2048:(
2045:n
2040:R
2038:(
2035:n
2032:M
2025:f
2021:S
2017:R
2012:S
2008:R
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1998:.
1993:X
1987:R
1983:f
1979:p
1975:X
1971:i
1967:)
1965:i
1963:(
1961:p
1957:p
1955:(
1953:f
1947:C
1943:R
1939:f
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1867:.
1859:x
1854:x
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1769:/
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1604:(
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896:,
893:a
866:,
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846:)
841:R
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814:(
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780:(
777:f
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764:b
761:(
758:f
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730:b
727:+
724:a
721:(
718:f
687:S
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678::
675:f
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506:p
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498:Z
486:p
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461:Q
448:p
429:p
424:Z
411:p
397:n
222:Z
218:1
214:/
209:Z
205:=
202:0
176:Z
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