1673:
is centrally primitive, we are done. If not, it is a sum of central orthogonal idempotents, which in turn are primitive or sums of more central idempotents, and so on. The problem that may occur is that this may continue without end, producing an infinite family of central orthogonal idempotents.
2768:
2652:
2812:
2739:
2705:
2679:
1683:" is a type of finiteness condition on the ring. It can be achieved in many ways, such as requiring the ring to be right
1183:
128:, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.
1179:
1015:
1008:
164:
145:
2531:
2094:
1821:
1004:
2700:, Mathematics and its Applications, vol. 575, Dordrecht: Kluwer Academic Publishers, pp. xii+380,
1045:
71:
2726:, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, pp. xx+385,
1506:
1304:
1230:
659:
643:
981:. Some authors use the term "idempotent ring" for this type of ring. In such a ring, multiplication is
2839:
1773:
1271:
1066:
593:
117:
956:
also do not have such idempotents, but for a different reason. The only idempotent contained in the
2799:, Algebras and Applications, vol. 1, Dordrecht: Kluwer Academic Publishers, pp. xii+371,
2478:
2249:
2042:
1843:
1765:
997:
188:
160:
121:
2329:
2304:
1977:
520:
47:
2786:
2808:
2764:
2735:
2701:
2675:
2648:
1186:
on left direct summands if and only if every set of pairwise orthogonal idempotents is finite.
897:
116:
For general rings, elements idempotent under multiplication are involved in decompositions of
110:
2800:
2774:
2727:
2486:
2482:
2127:
2117:
2074:
1172:
1149:
1077:
986:
982:
957:
760:
465:
168:
2822:
2749:
2715:
2689:
1776:
is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.
1411:
arises as left multiplication by a fixed ring element. With this modification of notation,
2818:
2778:
2745:
2711:
2685:
2644:
2510:
2282:
2266:
2262:
1684:
993:
945:
125:
2131:
1769:
2674:(2 ed.), Malabar, FL: Robert E. Krieger Publishing Co. Inc., pp. xviii+412,
305:
are not. This also demonstrates the decomposition properties described below: because
2833:
2151:
1059:
949:
781:
756:
398:
2538:
1408:
1055:
978:
912:
784:, and hence is a local ring, so right (and left) irreducible idempotents are local.
184:
17:
797:
that cannot be written as the sum of two nonzero orthogonal central idempotents.
663:
461:
106:
35:
31:
2804:
2756:
2731:
953:
727:
440:
67:
1011:
right (or every finitely generated left) ideal is generated by an idempotent.
2506:
2084:
2026:
1041:
666:
2255:
When the above partial order is restricted to the central idempotents of
1157:
97:
2664:
736:
is directly indecomposable, so local idempotents are also primitive.
2696:
Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2004),
1658:
is directly indecomposable as a ring if and only if the identity
1425:
as right modules if and only if there exists a unique idempotent
1681:
does not contain infinite sets of central orthogonal idempotents
2763:(Third ed.), Reading, Mass.: Addison-Wesley, p. 443,
2660:
2252:
of this partial order are precisely the primitive idempotents.
1182:
on right direct summands if and only if the ring satisfies the
1058:. Both of these types of rings are interesting even when they
1487:. Just as idempotents determine the direct decompositions of
2561:
199:. That is, each factor has two idempotents. So if there are
191:, so it is clear that the factors' only idempotents will be
476:
A partial list of important types of idempotents includes:
2526:
is not invertible are not difficult to find. The element
2041:
Lifting idempotents also has major consequences for the
2269:
structure, can be given. For two central idempotents
675:
is a primitive idempotent if it cannot be written as
2795:
Polcino Milies, César; Sehgal, Sudarshan K. (2002),
2198:the largest idempotent. For orthogonal idempotents
1665:Working inductively, one can attempt to decompose
1264:have an important connection to decomposition of
1669:into a sum of centrally primitive elements. If
2639:Anderson, Frank Wylie; Fuller, Kent R (1992),
2597:
2573:
1471:is a central idempotent, then the corner ring
1229:. The corner ring arises naturally since the
1207:is again a ring, with multiplicative identity
27:In mathematics, element that equals its square
2008:are orthogonal idempotents, corresponding to
1612:. So in particular, every central idempotent
8:
2154:on the idempotents of a ring as follows: if
1881:gives rise to two involutions of the module
1321:if and only if there is a unique idempotent
996:if and only if every right (or every left)
2562:Hazewinkel, Gubareni & Kirichenko 2004
1790:is an idempotent of the endomorphism ring
1582:that are pairwise orthogonal and have sum
1535:, then the identity elements of the rings
1363:is directly indecomposable if and only if
1040:is generated by an idempotent is called a
105:. For example, an idempotent element of a
2083:-module. Idempotents always lift modulo
2468:
1556:. Conversely, given central idempotents
1552:, pairwise orthogonal, and their sum is
1493:as a module, the central idempotents of
1390:(assumed unital), the endomorphism ring
429:. For example, this could be applied to
2554:
2498:
2029:to involutions in a one-to-one manner.
2025:is invertible, the idempotent elements
1736:a centrally primitive idempotent, then
1481:is a ring with multiplicative identity
2724:A first course in noncommutative rings
1073:. Such rings need not be commutative.
1044:. If the condition only holds for all
701:are nonzero orthogonal idempotents in
1761:, each of which is ring irreducible.
456:Idempotents in split-quaternion rings
7:
1742:is a direct sum of the corner rings
1630:as a direct sum of the corner rings
1065:A ring in which all idempotents are
977:elements are idempotent is called a
297:are idempotents of this ring, while
2698:Algebras, rings and modules. Vol. 1
2621:
2609:
2585:
2126:. Yet another characterization of
1899:represents an arbitrary element of
791:idempotent is a central idempotent
212:We can check this for the integers
1142:a local idempotent if and only if
968:Rings characterized by idempotents
25:
2530:is not invertible in any ring of
1624:gives rise to a decomposition of
1088:are the only central idempotents.
609:refers to either of the elements
187:. Now each of these factors is a
74:then, one can also conclude that
70:under the ring's multiplication.
2393:, and the join and meet satisfy
2368:The ordering now becomes simply
2220:is also idempotent, and we have
1972:This process can be reversed if
1854:is the identity endomorphism of
1499:determine the decompositions of
1456:. Thus every direct summand of
309:, there is a ring decomposition
2641:Rings and Categories of Modules
2105:Lifting is most important when
1592:is the direct sum of the rings
1515:is the direct sum of the rings
1462:is generated by an idempotent.
357:the multiplicative identity is
339:the multiplicative identity is
2797:An introduction to group rings
2190:. With respect to this order,
2134:whose idempotents lift modulo
2051:. All idempotents lift modulo
1893:as a left or right module. If
1875:and its associated involution
1060:lack a multiplicative identity
1000:is generated by an idempotent.
952:do not have such idempotents.
617:, which are always idempotent.
1:
1943:can also be viewed as a left
985:and every element is its own
167:, this ring factors into the
1371:are the only idempotents in
741:right irreducible idempotent
652:-module; that is, such that
2021:. Thus for a ring in which
1546:are central idempotents in
1165:ring if all idempotents of
1054:, then the ring is a right
905:Any non-trivial idempotent
368:Quotient of polynomial ring
124:properties of the ring. In
2856:
2788:Linear Associative Algebra
2598:Anderson & Fuller 1992
2574:Anderson & Fuller 1992
2166:are idempotents, we write
1219:is often referred to as a
1195:is idempotent in the ring
1184:descending chain condition
829:if there is an idempotent
513:is idempotent in the ring
66:. That is, the element is
2805:10.1007/978-94-010-0405-3
2785:Peirce, Benjamin (1870),
2732:10.1007/978-1-4419-8616-0
2672:von Neumann regular rings
1911:can be viewed as a right
1780:Relation with involutions
1180:ascending chain condition
472:Types of ring idempotents
281:From these computations,
165:Chinese remainder theorem
2670:Goodearl, K. R. (1991),
2485:, then the lattice is a
1804:, then the endomorphism
1662:is centrally primitive.
630:is a nonzero idempotent
171:of integers modulo
146:ring of integers modulo
1992:is an involution, then
1887:, depending on viewing
1652:. As a result, a ring
893:separability idempotent
234:has two prime factors (
205:factors, there will be
2146:Lattice of idempotents
1863:An idempotent element
1687:. If a decomposition
1254:Role in decompositions
464:of idempotents in the
2471:, p. 99 that if
2265:structure, or even a
2057:if and only if every
1949:-module homomorphism
1917:-module homomorphism
1305:ring of endomorphisms
1231:ring of endomorphisms
1178:A ring satisfies the
1014:A ring for which the
1007:if and only if every
810:in the quotient ring
730:. This implies that
144:One may consider the
2643:, Berlin, New York:
2194:is the smallest and
2087:and rings for which
1774:Peirce decomposition
1766:associative algebras
1155:A ring is called an
1078:directly irreducible
622:primitive idempotent
2722:Lam, T. Y. (2001),
2509:were introduced by
2479:von Neumann regular
1844:module homomorphism
1258:The idempotents of
1005:von Neumann regular
944:). This shows that
789:centrally primitive
423:has the idempotent
120:, and connected to
18:Centrally primitive
2467:. It is shown in
2100:-adically complete
2063:direct summand of
1978:invertible element
1772:over a field, the
1097:can be written as
1009:finitely generated
934:being zero, where
607:trivial idempotent
562:central idempotent
276:5 ≡ 25 ≡ 1 (mod 6)
271:4 ≡ 16 ≡ 4 (mod 6)
266:3 ≡ 9 ≡ 3 (mod 6)
40:idempotent element
2770:978-0-201-55540-0
2654:978-0-387-97845-1
2537:, which includes
2150:One may define a
2130:is that they are
2128:semiperfect rings
1725:exists with each
1380:In the case when
1357:. Clearly then,
898:Separable algebra
743:is an idempotent
714:is an idempotent
307:3 + 4 ≡ 1 (mod 6)
261:2 ≡ 4 ≡ 4 (mod 6)
256:1 ≡ 1 ≡ 1 (mod 6)
251:0 ≡ 0 ≡ 0 (mod 6)
242:) it should have
111:idempotent matrix
96:for any positive
16:(Redirected from
2847:
2825:
2791:
2781:
2752:
2718:
2692:
2657:
2625:
2619:
2613:
2607:
2601:
2595:
2589:
2583:
2577:
2571:
2565:
2559:
2542:
2536:
2529:
2525:
2520:
2514:
2503:
2487:complete lattice
2476:
2466:
2461:
2448:
2434:
2422:
2418:
2404:
2392:
2388:
2377:
2364:
2323:
2298:
2280:
2274:
2260:
2247:
2233:
2219:
2209:
2203:
2197:
2193:
2189:
2175:
2165:
2159:
2141:
2125:
2118:Jacobson radical
2115:
2099:
2092:
2082:
2075:projective cover
2072:
2062:
2056:
2048:
2024:
2020:
2013:
2007:
1999:
1991:
1985:
1975:
1968:
1958:
1948:
1942:
1936:
1926:
1916:
1910:
1904:
1898:
1892:
1886:
1880:
1874:
1868:
1859:
1853:
1852:
1841:
1835:
1829:
1819:
1813:
1803:
1789:
1760:
1741:
1735:
1724:
1679:
1672:
1668:
1661:
1657:
1651:
1635:
1629:
1623:
1617:
1611:
1600:
1591:
1585:
1581:
1575:
1564:
1555:
1551:
1545:
1534:
1523:
1514:
1504:
1498:
1492:
1486:
1480:
1470:
1461:
1455:
1440:
1430:
1424:
1406:
1389:
1376:
1370:
1366:
1362:
1356:
1342:
1332:
1326:
1320:
1302:
1285:
1279:
1269:
1263:
1248:
1228:
1218:
1212:
1206:
1200:
1194:
1173:Jacobson radical
1171:lift modulo the
1170:
1150:semiperfect ring
1147:
1141:
1130:
1096:
1087:
1083:
1053:
1039:
1033:
1027:
987:additive inverse
973:A ring in which
963:
958:Jacobson radical
946:integral domains
943:
933:
927:
921:
910:
887:
873:
867:
858:
840:
834:
827:
819:
809:
796:
779:
754:
748:
735:
725:
719:
712:local idempotent
706:
700:
694:
688:
674:
669:. Equivalently,
657:
651:
641:
635:
629:
616:
612:
601:
591:
585:
579:
573:
559:
553:
544:
538:
532:
518:
512:
506:
491:
485:
480:Two idempotents
466:split-quaternion
451:
438:
428:
419:
413:
396:
394:
387:
377:
363:
356:
345:
338:
327:
308:
304:
300:
296:
292:
288:
284:
277:
272:
267:
262:
257:
252:
245:
241:
237:
233:
229:
215:
208:
204:
198:
194:
182:
176:
169:product of rings
158:
151:
109:is precisely an
104:
95:
65:
55:
21:
2855:
2854:
2850:
2849:
2848:
2846:
2845:
2844:
2830:
2829:
2828:
2815:
2794:
2784:
2771:
2755:
2742:
2721:
2708:
2695:
2682:
2669:
2655:
2645:Springer-Verlag
2638:
2634:
2629:
2628:
2620:
2616:
2608:
2604:
2596:
2592:
2584:
2580:
2572:
2568:
2560:
2556:
2551:
2546:
2545:
2534:
2527:
2523:
2522:Rings in which
2521:
2517:
2511:Benjamin Peirce
2505:Idempotent and
2504:
2500:
2495:
2472:
2457:
2444:
2430:
2424:
2414:
2400:
2394:
2384:
2379:
2378:if and only if
2369:
2336:
2311:
2289:
2276:
2270:
2267:Boolean algebra
2256:
2235:
2221:
2211:
2205:
2199:
2195:
2191:
2177:
2176:if and only if
2167:
2161:
2155:
2148:
2135:
2132:semilocal rings
2121:
2106:
2095:
2088:
2078:
2064:
2058:
2052:
2044:
2039:
2022:
2015:
2009:
2001:
1993:
1987:
1981:
1973:
1960:
1950:
1944:
1938:
1928:
1918:
1912:
1906:
1900:
1894:
1888:
1882:
1876:
1870:
1864:
1855:
1848:
1847:
1837:
1831:
1825:
1815:
1805:
1797:
1791:
1785:
1782:
1770:Jordan algebras
1759:
1751:
1743:
1737:
1734:
1726:
1720:
1708:
1698:
1688:
1675:
1674:The condition "
1670:
1666:
1659:
1653:
1637:
1631:
1625:
1619:
1613:
1610:
1602:
1599:
1593:
1587:
1583:
1577:
1574:
1566:
1563:
1557:
1553:
1547:
1544:
1536:
1533:
1525:
1522:
1516:
1510:
1500:
1494:
1488:
1482:
1472:
1466:
1457:
1442:
1432:
1426:
1412:
1397:
1391:
1381:
1372:
1368:
1364:
1358:
1344:
1334:
1328:
1322:
1308:
1296:
1287:
1281:
1275:
1265:
1259:
1256:
1239:
1233:
1224:
1214:
1208:
1202:
1196:
1190:
1166:
1143:
1140:
1132:
1126:
1114:
1104:
1098:
1092:
1085:
1081:
1080:if and only if
1049:
1035:
1029:
1018:
970:
961:
935:
929:
923:
916:
906:
879:
876:full idempotent
869:
863:
842:
836:
830:
823:
811:
801:
792:
770:
764:
750:
744:
731:
721:
715:
702:
696:
690:
676:
670:
653:
647:
637:
631:
625:
614:
610:
597:
587:
581:
575:
565:
555:
549:
545:are orthogonal.
540:
534:
524:
514:
508:
497:
487:
481:
474:
458:
443:
430:
424:
409:
404:
390:
389:
379:
378:and an element
373:
370:
358:
347:
340:
329:
310:
306:
302:
298:
294:
290:
286:
282:
275:
270:
265:
260:
255:
250:
243:
239:
235:
231:
217:
213:
206:
200:
196:
192:
178:
172:
154:
147:
142:
134:
126:Boolean algebra
100:
75:
57:
51:
28:
23:
22:
15:
12:
11:
5:
2853:
2851:
2843:
2842:
2832:
2831:
2827:
2826:
2813:
2792:
2782:
2769:
2753:
2740:
2719:
2706:
2693:
2680:
2667:
2658:
2653:
2635:
2633:
2630:
2627:
2626:
2614:
2602:
2590:
2578:
2566:
2553:
2552:
2550:
2547:
2544:
2543:
2532:characteristic
2515:
2497:
2496:
2494:
2491:
2483:self-injective
2366:
2365:
2326:
2325:
2301:
2300:
2147:
2144:
2038:
2031:
1793:
1781:
1778:
1755:
1747:
1730:
1716:
1706:
1696:
1606:
1597:
1570:
1561:
1540:
1529:
1520:
1393:
1292:
1255:
1252:
1251:
1250:
1235:
1187:
1176:
1153:
1136:
1122:
1112:
1102:
1089:
1074:
1063:
1012:
1001:
990:
969:
966:
950:division rings
903:
902:
889:
862:An idempotent
860:
800:An idempotent
798:
785:
766:
737:
708:
644:indecomposable
618:
603:
586:, that is, if
548:An idempotent
546:
523:), then so is
473:
470:
457:
454:
421:
420:
369:
366:
279:
278:
273:
268:
263:
258:
253:
141:
135:
133:
130:
50:is an element
34:, a branch of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2852:
2841:
2838:
2837:
2835:
2824:
2820:
2816:
2814:1-4020-0238-6
2810:
2806:
2802:
2798:
2793:
2790:
2789:
2783:
2780:
2776:
2772:
2766:
2762:
2758:
2754:
2751:
2747:
2743:
2741:0-387-95183-0
2737:
2733:
2729:
2725:
2720:
2717:
2713:
2709:
2707:1-4020-2690-0
2703:
2699:
2694:
2691:
2687:
2683:
2681:0-89464-632-X
2677:
2673:
2668:
2666:
2662:
2659:
2656:
2650:
2646:
2642:
2637:
2636:
2631:
2624:, p. 323
2623:
2618:
2615:
2611:
2606:
2603:
2599:
2594:
2591:
2587:
2582:
2579:
2575:
2570:
2567:
2563:
2558:
2555:
2548:
2540:
2539:Boolean rings
2533:
2519:
2516:
2512:
2508:
2502:
2499:
2492:
2490:
2488:
2484:
2480:
2475:
2470:
2469:Goodearl 1991
2464:
2460:
2455:
2451:
2447:
2442:
2438:
2433:
2428:
2421:
2417:
2412:
2408:
2403:
2398:
2391:
2387:
2382:
2376:
2372:
2363:
2359:
2355:
2351:
2347:
2343:
2339:
2335:
2334:
2333:
2331:
2322:
2318:
2314:
2310:
2309:
2308:
2306:
2297:
2293:
2288:
2287:
2286:
2284:
2279:
2273:
2268:
2264:
2259:
2253:
2251:
2246:
2242:
2238:
2232:
2228:
2224:
2218:
2214:
2208:
2202:
2188:
2184:
2180:
2174:
2170:
2164:
2158:
2153:
2152:partial order
2145:
2143:
2139:
2133:
2129:
2124:
2119:
2113:
2109:
2103:
2101:
2098:
2091:
2086:
2081:
2076:
2071:
2067:
2061:
2055:
2050:
2047:
2036:
2032:
2030:
2028:
2019:
2012:
2005:
1997:
1990:
1984:
1979:
1970:
1967:
1963:
1957:
1953:
1947:
1941:
1935:
1931:
1925:
1921:
1915:
1909:
1903:
1897:
1891:
1885:
1879:
1873:
1867:
1861:
1858:
1851:
1845:
1840:
1834:
1828:
1823:
1818:
1812:
1808:
1801:
1796:
1788:
1779:
1777:
1775:
1771:
1767:
1762:
1758:
1754:
1750:
1746:
1740:
1733:
1729:
1723:
1719:
1715:
1711:
1705:
1701:
1695:
1691:
1686:
1682:
1678:
1663:
1656:
1649:
1645:
1641:
1634:
1628:
1622:
1616:
1609:
1605:
1596:
1590:
1580:
1573:
1569:
1560:
1550:
1543:
1539:
1532:
1528:
1519:
1513:
1509:of rings. If
1508:
1503:
1497:
1491:
1485:
1479:
1475:
1469:
1463:
1460:
1454:
1450:
1446:
1439:
1435:
1429:
1423:
1419:
1415:
1410:
1407:, where each
1405:
1401:
1396:
1388:
1384:
1378:
1375:
1361:
1355:
1351:
1347:
1341:
1337:
1331:
1325:
1319:
1315:
1311:
1306:
1300:
1295:
1290:
1284:
1278:
1273:
1268:
1262:
1253:
1247:
1243:
1238:
1232:
1227:
1222:
1217:
1211:
1205:
1199:
1193:
1188:
1185:
1181:
1177:
1174:
1169:
1164:
1160:
1159:
1154:
1151:
1146:
1139:
1135:
1129:
1125:
1121:
1117:
1111:
1107:
1101:
1095:
1090:
1079:
1075:
1072:
1069:is called an
1068:
1064:
1061:
1057:
1052:
1047:
1043:
1038:
1032:
1028:every subset
1025:
1021:
1017:
1013:
1010:
1006:
1002:
999:
995:
991:
988:
984:
980:
976:
972:
971:
967:
965:
960:of a ring is
959:
955:
951:
947:
942:
938:
932:
926:
922:with neither
919:
914:
909:
900:
899:
894:
890:
886:
882:
877:
872:
866:
861:
857:
853:
849:
845:
839:
833:
828:
826:
818:
814:
808:
804:
799:
795:
790:
786:
783:
782:division ring
778:
774:
769:
762:
761:Schur's lemma
758:
757:simple module
753:
747:
742:
738:
734:
729:
724:
718:
713:
709:
705:
699:
693:
687:
683:
679:
673:
668:
665:
661:
656:
650:
645:
640:
634:
628:
623:
619:
608:
604:
600:
595:
590:
584:
578:
572:
568:
563:
558:
552:
547:
543:
537:
531:
527:
522:
517:
511:
504:
500:
495:
490:
484:
479:
478:
477:
471:
469:
467:
463:
455:
453:
450:
446:
442:
437:
433:
427:
417:
412:
407:
403:
402:
401:
400:
399:quotient ring
393:
386:
382:
376:
372:Given a ring
367:
365:
362:
355:
351:
344:
337:
333:
326:
322:
318:
314:
274:
269:
264:
259:
254:
249:
248:
247:
246:idempotents.
228:
224:
220:
210:
209:idempotents.
203:
190:
186:
181:
175:
170:
166:
162:
157:
152:
150:
140:
137:Quotients of
136:
131:
129:
127:
123:
119:
114:
112:
108:
103:
99:
94:
90:
86:
82:
78:
73:
69:
64:
60:
54:
49:
45:
41:
37:
33:
19:
2796:
2787:
2760:
2723:
2697:
2671:
2640:
2617:
2605:
2593:
2581:
2569:
2557:
2518:
2501:
2473:
2462:
2458:
2453:
2449:
2445:
2440:
2436:
2431:
2426:
2419:
2415:
2410:
2406:
2401:
2396:
2389:
2385:
2380:
2374:
2370:
2367:
2361:
2357:
2353:
2349:
2345:
2341:
2337:
2332:is given by
2327:
2320:
2316:
2312:
2307:is given by
2302:
2295:
2291:
2285:is given by
2277:
2271:
2257:
2254:
2244:
2240:
2236:
2230:
2226:
2222:
2216:
2212:
2206:
2200:
2186:
2182:
2178:
2172:
2168:
2162:
2156:
2149:
2137:
2122:
2111:
2107:
2104:
2096:
2089:
2079:
2069:
2065:
2059:
2053:
2045:
2043:category of
2040:
2034:
2033:Category of
2017:
2010:
2003:
1995:
1988:
1982:
1971:
1965:
1961:
1955:
1951:
1945:
1939:
1933:
1929:
1923:
1919:
1913:
1907:
1901:
1895:
1889:
1883:
1877:
1871:
1865:
1862:
1856:
1849:
1838:
1832:
1826:
1816:
1810:
1806:
1799:
1794:
1786:
1783:
1763:
1756:
1752:
1748:
1744:
1738:
1731:
1727:
1721:
1717:
1713:
1709:
1703:
1699:
1693:
1689:
1680:
1676:
1664:
1654:
1647:
1643:
1639:
1632:
1626:
1620:
1614:
1607:
1603:
1594:
1588:
1578:
1571:
1567:
1558:
1548:
1541:
1537:
1530:
1526:
1517:
1511:
1501:
1495:
1489:
1483:
1477:
1473:
1467:
1464:
1458:
1452:
1448:
1444:
1437:
1433:
1427:
1421:
1417:
1413:
1409:endomorphism
1403:
1399:
1394:
1386:
1382:
1379:
1373:
1359:
1353:
1349:
1345:
1339:
1335:
1329:
1323:
1317:
1313:
1309:
1298:
1293:
1288:
1286:-module and
1282:
1276:
1266:
1260:
1257:
1245:
1241:
1236:
1225:
1220:
1215:
1213:. The ring
1209:
1203:
1197:
1191:
1167:
1162:
1156:
1144:
1137:
1133:
1127:
1123:
1119:
1115:
1109:
1105:
1099:
1093:
1071:abelian ring
1070:
1056:Rickart ring
1050:
1036:
1030:
1023:
1019:
979:Boolean ring
974:
940:
936:
930:
924:
917:
913:zero divisor
907:
904:
896:
892:
884:
880:
875:
874:is called a
870:
864:
855:
851:
847:
843:
837:
831:
824:
822:lift modulo
821:
816:
812:
806:
802:
793:
788:
776:
772:
767:
751:
745:
740:
732:
722:
716:
711:
703:
697:
691:
685:
681:
677:
671:
654:
648:
638:
632:
626:
621:
606:
598:
588:
582:
576:
570:
566:
561:
560:is called a
556:
550:
541:
535:
533:; moreover,
529:
525:
515:
509:
502:
498:
493:
488:
482:
475:
459:
448:
444:
435:
431:
425:
422:
415:
410:
405:
391:
384:
380:
374:
371:
360:
353:
349:
342:
335:
331:
324:
320:
316:
312:
280:
226:
222:
218:
211:
201:
179:
173:
155:
148:
143:
138:
115:
101:
92:
88:
84:
80:
76:
62:
58:
52:
43:
39:
29:
2840:Ring theory
2757:Lang, Serge
2564:, p. 2
1830:. That is,
1221:corner ring
1048:subsets of
1016:annihilator
983:commutative
954:Local rings
820:is said to
646:as a right
492:are called
462:hyperboloid
460:There is a
122:homological
107:matrix ring
72:Inductively
36:mathematics
32:ring theory
2779:0848.13001
2661:idempotent
2632:References
2576:, p. 69–72
2481:and right
2283:complement
2085:nil ideals
2027:correspond
1846:such that
1822:involution
1685:Noetherian
1507:direct sum
1431:such that
1333:such that
1131:with each
1076:A ring is
1003:A ring is
994:semisimple
992:A ring is
841:such that
749:for which
728:local ring
720:such that
667:submodules
660:direct sum
636:such that
624:of a ring
592:is in the
494:orthogonal
441:polynomial
388:such that
214:mod 6
161:squarefree
68:idempotent
56:such that
44:idempotent
42:or simply
2549:Citations
2507:nilpotent
1046:singleton
1042:Baer ring
915:(because
658:is not a
439:, or any
230:. Since
163:. By the
2834:Category
2759:(1993),
2622:Lam 2001
2612:, p. 336
2610:Lam 2001
2600:, p. 302
2588:, p. 326
2586:Lam 2001
2513:in 1870.
2328:and the
2049:-modules
2037:-modules
1959:, where
1927:so that
1820:-module
1712:⊕ ... ⊕
1163:Lift/rad
1158:SBI ring
1118:⊕ ... ⊕
689:, where
574:for all
177:, where
153:, where
132:Examples
91:= ... =
2823:1896125
2761:Algebra
2750:1838439
2716:2106764
2690:1150975
2263:lattice
2248:. The
1809:= 1 − 2
1601:, ...,
1586:, then
1565:, ...,
1524:, ...,
1348:= (1 −
1307:, then
1303:is its
1272:modules
1201:, then
1091:A ring
1067:central
664:nonzero
662:of two
346:and in
118:modules
98:integer
2821:
2811:
2777:
2767:
2748:
2738:
2714:
2704:
2688:
2678:
2665:FOLDOC
2651:
2294:= 1 −
2281:, the
2116:, the
2077:as an
2073:has a
2002:2(1 +
1994:2(1 −
1976:is an
1836:is an
1814:is an
1280:is an
1274:. If
939:= 1 −
895:; see
759:. By
594:centre
528:= 1 −
519:(with
468:ring.
397:, the
328:. In
293:, and
2493:Notes
2344:= ¬(¬
2250:atoms
1986:: if
1937:, or
1646:(1 −
1638:(1 −
1505:as a
1443:(1 −
1291:= End
1148:is a
1022:.Ann(
998:ideal
911:is a
780:is a
755:is a
726:is a
521:unity
507:. If
359:4 + 6
341:3 + 6
189:field
185:prime
46:of a
38:, an
2809:ISBN
2765:ISBN
2736:ISBN
2702:ISBN
2676:ISBN
2649:ISBN
2423:and
2352:) =
2330:join
2305:meet
2303:the
2275:and
2261:, a
2234:and
2204:and
2160:and
2110:= J(
2016:1 −
2014:and
2000:and
1764:For
1636:and
1441:and
1402:) =
1367:and
1343:and
1244:) ≅
1084:and
948:and
928:nor
775:) =
695:and
613:and
539:and
486:and
301:and
238:and
195:and
48:ring
2801:doi
2775:Zbl
2728:doi
2663:at
2477:is
2452:= (
2348:∧ ¬
2120:of
2093:is
1980:of
1962:rff
1930:ffr
1869:of
1824:of
1792:End
1784:If
1768:or
1633:aRa
1618:in
1576:in
1474:aRa
1465:If
1392:End
1327:in
1246:aRa
1234:End
1223:of
1216:aRa
1204:aRa
1189:If
1161:or
1034:of
975:all
920:= 0
881:RaR
878:if
868:of
835:in
777:aRa
765:End
723:aRa
642:is
596:of
580:in
564:if
554:in
505:= 0
496:if
408:/ (
395:≠ 0
352:/ 6
334:/ 6
323:/ 6
319:⊕ 4
315:/ 6
225:/ 6
183:is
159:is
30:In
2836::
2819:MR
2817:,
2807:,
2773:,
2746:MR
2744:,
2734:,
2712:MR
2710:,
2686:MR
2684:,
2647:,
2489:.
2456:)(
2454:eR
2443:∩
2441:eR
2439:=
2429:∧
2413:+
2411:eR
2409:=
2399:∨
2383:⊆
2381:eR
2373:≤
2362:ef
2360:−
2356:+
2340:∨
2321:ef
2319:=
2315:∧
2243:+
2239:≤
2229:+
2225:≤
2215:+
2210:,
2185:=
2183:ba
2181:=
2179:ab
2171:≤
2142:.
2136:J(
2102:.
1969:.
1964:=
1956:rf
1954:↦
1932:=
1924:fr
1922:↦
1905:,
1860:.
1753:Rc
1702:⊕
1692:=
1604:Ra
1595:Ra
1478:Ra
1476:=
1451:=
1436:=
1434:eR
1420:=
1416:⊕
1385:=
1377:.
1340:eM
1338:=
1316:=
1312:⊕
1242:aR
1108:⊕
964:.
918:ab
891:A
883:=
854:+
850:=
846:+
815:/
805:+
787:A
773:aR
763:,
752:aR
739:A
733:aR
710:A
684:+
680:=
655:aR
639:aR
620:A
605:A
571:xa
569:=
567:ax
503:ba
501:=
499:ab
452:.
447:∈
434:∈
414:−
383:∈
364:.
289:,
285:,
221:=
216:,
113:.
87:=
83:=
79:=
61:=
2803::
2730::
2541:.
2535:2
2528:2
2524:2
2474:R
2465:)
2463:R
2459:f
2450:R
2446:f
2437:R
2435:)
2432:f
2427:e
2425:(
2420:R
2416:f
2407:R
2405:)
2402:f
2397:e
2395:(
2390:R
2386:f
2375:f
2371:e
2358:f
2354:e
2350:f
2346:e
2342:f
2338:e
2324:.
2317:f
2313:e
2299:,
2296:e
2292:e
2290:¬
2278:f
2272:e
2258:R
2245:b
2241:a
2237:b
2231:b
2227:a
2223:a
2217:b
2213:a
2207:b
2201:a
2196:1
2192:0
2187:a
2173:b
2169:a
2163:b
2157:a
2140:)
2138:R
2123:R
2114:)
2112:R
2108:I
2097:I
2090:R
2080:R
2070:I
2068:/
2066:R
2060:R
2054:I
2046:R
2035:R
2023:2
2018:a
2011:a
2006:)
2004:b
1998:)
1996:b
1989:b
1983:R
1974:2
1966:r
1952:r
1946:R
1940:f
1934:r
1920:r
1914:R
1908:f
1902:R
1896:r
1890:R
1884:R
1878:f
1872:R
1866:a
1857:M
1850:f
1842:-
1839:R
1833:f
1827:M
1817:R
1811:a
1807:f
1802:)
1800:M
1798:(
1795:R
1787:a
1757:i
1749:i
1745:c
1739:R
1732:i
1728:c
1722:R
1718:n
1714:c
1710:R
1707:2
1704:c
1700:R
1697:1
1694:c
1690:R
1677:R
1671:1
1667:1
1660:1
1655:R
1650:)
1648:a
1644:R
1642:)
1640:a
1627:R
1621:R
1615:a
1608:n
1598:1
1589:R
1584:1
1579:R
1572:n
1568:a
1562:1
1559:a
1554:1
1549:R
1542:i
1538:R
1531:n
1527:R
1521:1
1518:R
1512:R
1502:R
1496:R
1490:R
1484:a
1468:a
1459:R
1453:B
1449:R
1447:)
1445:e
1438:A
1428:e
1422:R
1418:B
1414:A
1404:R
1400:R
1398:(
1395:R
1387:R
1383:M
1374:E
1369:1
1365:0
1360:M
1354:M
1352:)
1350:e
1346:B
1336:A
1330:E
1324:e
1318:M
1314:B
1310:A
1301:)
1299:M
1297:(
1294:R
1289:E
1283:R
1277:M
1270:-
1267:R
1261:R
1249:.
1240:(
1237:R
1226:R
1210:a
1198:R
1192:a
1175:.
1168:R
1152:.
1145:R
1138:i
1134:e
1128:R
1124:n
1120:e
1116:R
1113:2
1110:e
1106:R
1103:1
1100:e
1094:R
1086:1
1082:0
1062:.
1051:R
1037:R
1031:S
1026:)
1024:S
1020:r
989:.
962:0
941:a
937:b
931:b
925:a
908:a
901:.
888:.
885:R
871:R
865:a
859:.
856:I
852:a
848:I
844:b
838:R
832:b
825:I
817:I
813:R
807:I
803:a
794:a
771:(
768:R
746:a
717:a
707:.
704:R
698:f
692:e
686:f
682:e
678:a
672:a
649:R
633:a
627:R
615:1
611:0
602:.
599:R
589:a
583:R
577:x
557:R
551:a
542:b
536:a
530:a
526:b
516:R
510:a
489:b
483:a
449:k
445:f
436:Z
432:x
426:f
418:)
416:f
411:f
406:R
392:f
385:R
381:f
375:R
361:Z
354:Z
350:Z
348:4
343:Z
336:Z
332:Z
330:3
325:Z
321:Z
317:Z
313:Z
311:3
303:5
299:2
295:4
291:3
287:1
283:0
244:2
240:3
236:2
232:6
227:Z
223:Z
219:R
207:2
202:m
197:1
193:0
180:p
174:p
156:n
149:n
139:Z
102:n
93:a
89:a
85:a
81:a
77:a
63:a
59:a
53:a
20:)
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