185:
2300:-projective. Usually relative properties of subrings or ring extensions, such as the notion of separable extension, serve to promote theorems that say that the over-ring shares a property of the subring. For example, a separable extension
2133:
1199:
479:
646:
1618:
104:
1102:
1344:
896:
1792:
2531:
1245:
239:
1847:
698:
1879:
543:
829:
1938:
1502:
793:
731:
926:
303:
89:
1704:
1669:
1008:
390:
367:
347:
323:
277:
1469:
2024:
2683:
2571:
2510:
401:
1107:
570:
2654:
2621:
2138:
An algebra is strongly separable if and only if its trace form is nondegenerate, thus making the algebra into a particular kind of
988:
1550:
180:{\displaystyle {\begin{array}{rccc}\mu :&A\otimes _{K}A&\to &A\\&a\otimes b&\mapsto &ab\end{array}}}
1013:
964:
1274:
834:
1404:
1752:
2162:
1204:
199:
191:
1396:
1373:
2365:
has finite representation type, the restricted module is uniquely a direct sum of multiples of finitely many
1813:
2646:
665:
1881:. The converse can be proven by a judicious use of the separability idempotent (similarly to the proof of
1852:
2366:
1889:
502:
1415:-algebras are finite products of matrix algebras over finite-dimensional division algebras over field
1882:
1730:
1718:
1641:
1525:
757:
746:
326:
2195:
1380:
1362:
968:
802:
40:
36:
2601:
1895:
2588:
Hirata, H.; Sugano, K. (1966), "On semisimple and separable extensions of noncommutative rings",
1428:
60:
32:
1474:
944:
entry, which is 1. In particular, this shows that separability idempotents need not be unique.
109:
2718:
2697:
2679:
2650:
2617:
2567:
2506:
2219:
2165:
2143:
2139:
1637:
1351:
1256:
929:
762:
2347:
to the characteristic. The separability condition above will imply every finitely generated
703:
2660:
2609:
2527:
2516:
1376:
901:
282:
68:
56:
17:
2693:
2631:
2581:
2555:
1682:
1647:
2689:
2664:
2627:
2608:, University Lecture Series, vol. 14, Providence, RI: American Mathematical Society,
2577:
2551:
2532:
On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings
2520:
2502:
2260:
is that it is left or right semisimple extension: a short exact sequence of left or right
953:
2678:. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press.
2671:
2230:
A theorem in the area is that of J. Cuadra that a separable Hopf–Galois extension
2147:
1369:
993:
375:
352:
332:
308:
262:
2712:
2638:
2381:
is. The converse is proven by a similar argument noting that every subgroup algebra
2128:{\displaystyle e=\sum _{i=1}^{n}x_{i}\otimes y_{i}=\sum _{i=1}^{n}y_{i}\otimes x_{i}}
1963:. Examples of separable extensions are many including first separable algebras where
1446:
1392:
2281:. In terms of G. Hochschild's relative homological algebra, one says that all
2328:
1521:
1400:
52:
2539:
2369:, which induce to a finite number of constituent indecomposable modules of which
1749:-bimodules. Indeed, this condition is necessary since the multiplication mapping
2335:-groups, then note that the group algebra is a separable extension of its Sylow
2207:
1676:
24:
2358:
1537:
658:
2701:
1423:
is a perfect field, there is no difference between a separable algebra over
2501:. Lecture Notes in Mathematics. Vol. 181. Berlin-Heidelberg-New York:
1709:
Separable algebras can also be characterized by means of split extensions:
1435:. It can be shown by a generalized theorem of Maschke that an associative
1387:. In particular: Every separable algebra is itself finite-dimensional. If
253:
2613:
2344:
474:{\displaystyle p:=\sigma (1)=\sum a_{i}\otimes b_{i}\in A\otimes _{K}A}
1368:
can be classified as follows: they are the same as finite products of
2645:, London Mathematical Society Monographs. New Series, vol. 28,
1194:{\textstyle \sum _{i=0}^{n-1}a^{i}\otimes _{K}{\frac {b_{i}}{p'(a)}}}
1885:, applying its components within and without the splitting maps).
1628:
There are several equivalent definitions of separable algebras. A
641:{\displaystyle \sum aa_{i}\otimes b_{i}=\sum a_{i}\otimes b_{i}a.}
2315:
There is the celebrated Jans theorem that a finite group algebra
395:
It is useful to describe separability in terms of the element
2361:
to a direct summand in its restricted, induced module. But if
556:-bimodules is equivalent to the following requirement for any
2180:
Relation to formally unramified and formally étale extensions
1994:
different from 1, is a separable extension over the subring
1613:{\textstyle {\frac {1}{o(G)}}\sum _{g\in G}g\otimes g^{-1}}
2323:
is of finite representation type if and only if its Sylow
2312:
semisimple, which follows from the preceding discussion.
2142:
called a symmetric algebra (not to be confused with the
1201:. The tensorands are dual bases for the trace map: if
1816:
1755:
1553:
1477:
1449:
1277:
1207:
1110:
1016:
837:
2027:
1898:
1855:
1685:
1650:
996:
904:
805:
765:
706:
668:
573:
505:
404:
378:
355:
335:
311:
285:
265:
202:
107:
71:
548:and the condition that σ is a homomorphism of
488:is determined by this element. The condition that
2250:. A fundamental fact about a separable extension
2127:
2014:if there exists a separability idempotent that is
1932:
1873:
1841:
1786:
1698:
1663:
1612:
1496:
1463:
1346:. The trace map and its dual bases make explicit
1338:
1239:
1193:
1097:{\textstyle p(x)=(x-a)\sum _{i=0}^{n-1}b_{i}x^{i}}
1096:
1002:
920:
890:
823:
787:
725:
692:
640:
537:
473:
384:
361:
341:
317:
297:
271:
233:
179:
83:
2566:, Providence, RI: American Mathematical Society,
1547:-algebra. A separability idempotent is given by
1339:{\textstyle Tr(x)=\sum _{i=1}^{n}\sigma _{i}(x)}
891:{\textstyle \sum _{i=1}^{n}e_{ij}\otimes e_{ji}}
2540:"On separable algebras over a commutative ring"
2331:: the clearest proof is to note this fact for
662:, since regarded as an element of the algebra
1787:{\textstyle \mu :A\otimes _{K}A\rightarrow A}
1104:, then a separability idempotent is given by
8:
2448:
1802:-bimodule epimorphism, which is split as an
1624:Equivalent characterizations of separability
1240:{\textstyle \sigma _{1},\ldots ,\sigma _{n}}
1810:-bimodule map by the right inverse mapping
234:{\displaystyle \sigma :A\to A\otimes _{K}A}
2455:is commutative, the proof is simpler, see
1443:is separable if for every field extension
1361:More generally, separable algebras over a
2499:Separable algebras over commutative rings
2119:
2106:
2096:
2085:
2072:
2059:
2049:
2038:
2026:
1903:
1897:
1854:
1830:
1815:
1769:
1754:
1690:
1684:
1655:
1649:
1601:
1579:
1554:
1552:
1485:
1476:
1453:
1448:
1321:
1311:
1300:
1276:
1231:
1212:
1206:
1164:
1158:
1152:
1142:
1126:
1115:
1109:
1088:
1078:
1062:
1051:
1015:
995:
909:
903:
879:
863:
853:
842:
836:
804:
770:
764:
711:
705:
680:
679:
667:
626:
613:
597:
584:
572:
523:
513:
504:
462:
446:
433:
403:
377:
354:
334:
310:
284:
264:
222:
201:
127:
108:
106:
70:
2534:, Nagoya Math. J. Volume 9 (1955), 1–16.
1842:{\textstyle A\rightarrow A\otimes _{K}A}
831:, a separability idempotent is given by
2538:Endo, Shizuo; Watanabe, Yutaka (1967),
2456:
2405:
1671:in the usual way. Moreover, an algebra
932:which is 0 except for the entry in the
2676:An introduction to homological algebra
2424:
1971:= 1 times the ground field. Any ring
1794:arising in the definition above is a
1679:when considered as a right module of
1379:whose centers are finite-dimensional
693:{\displaystyle A\otimes A^{\rm {op}}}
7:
2606:New examples of Frobenius extensions
2480:
2468:
2436:
2412:
2377:is of finite representation type if
1874:{\displaystyle a\mapsto a\otimes 1}
2497:DeMeyer, F.; Ingraham, E. (1971).
2392:direct summand of a group algebra
2176:is a symmetric Frobenius algebra.
2010:A separable algebra is said to be
1675:is separable if and only if it is
1636:is separable if and only if it is
684:
681:
14:
538:{\displaystyle \sum a_{i}b_{i}=1}
2471:, Corollary 4.7.2, Theorem 8.3.6
1411:is separable, so that separable
55:of (unital, but not necessarily
2319:over a field of characteristic
2240:has finitely generated natural
2146:arising as the quotient of the
975:is separable as an associative
948:Separable algebras over a field
47:Definition and first properties
2006:Relation to Frobenius algebras
1927:
1909:
1859:
1820:
1778:
1569:
1563:
1333:
1327:
1290:
1284:
1185:
1179:
1044:
1032:
1026:
1020:
782:
776:
420:
414:
289:
212:
162:
138:
75:
1:
824:{\displaystyle 1\leq j\leq n}
484:The reason is that a section
35:. It is a generalization to
2544:Osaka Journal of Mathematics
1952:in any coefficient bimodule
1933:{\displaystyle H^{n}(R,S;M)}
1010:with irreducible polynomial
1967:is a separable algebra and
1888:Equivalently, the relative
1497:{\textstyle A\otimes _{K}L}
16:Not to be confused with an
2735:
1640:when considered as a left
1407:– then every extension of
1405:algebraically closed field
1381:separable field extensions
1263:, the trace mapping Tr of
244:that is a homomorphism of
98:if the multiplication map
15:
2562:Ford, Timothy J. (2017),
1741:-bimodules also split as
1427:and a finite-dimensional
1395:– for example a field of
741:For any commutative ring
41:separable field extension
2449:Endo & Watanabe 1967
2373:is a direct sum. Hence
2304:of a semisimple algebra
2202:is a finitely generated
2198:. The converse holds if
2194:of commutative rings is
2184:Any separable extension
1516:is commutative ring and
788:{\displaystyle M_{n}(R)}
2647:Oxford University Press
726:{\displaystyle p^{2}=p}
2206:-algebra. A separable
2129:
2101:
2054:
1934:
1875:
1843:
1788:
1700:
1665:
1614:
1498:
1465:
1419:. In other words, if
1340:
1316:
1241:
1195:
1137:
1098:
1073:
1004:
922:
921:{\displaystyle e_{ij}}
892:
858:
825:
789:
727:
694:
642:
539:
475:
386:
371:separable algebra over
363:
343:
319:
299:
298:{\displaystyle K\to A}
273:
235:
181:
85:
84:{\displaystyle K\to A}
2530:and Tadasi Nakayama,
2130:
2081:
2034:
1935:
1890:Hochschild cohomology
1876:
1844:
1789:
1719:short exact sequences
1701:
1699:{\displaystyle A^{e}}
1666:
1664:{\displaystyle A^{e}}
1615:
1499:
1466:
1341:
1296:
1242:
1196:
1111:
1099:
1047:
1005:
923:
893:
838:
826:
790:
728:
695:
643:
540:
476:
387:
364:
344:
320:
300:
274:
236:
182:
86:
2025:
1896:
1853:
1814:
1753:
1729:-bimodules that are
1706:in the usual way.
1683:
1648:
1551:
1475:
1447:
1275:
1205:
1108:
1014:
994:
902:
835:
803:
763:
704:
666:
571:
503:
402:
376:
353:
333:
309:
283:
263:
200:
105:
69:
37:associative algebras
2196:formally unramified
1998:generated by 1 and
1717:if and only if all
969:separable extension
279:is commutative and
39:of the notion of a
2672:Weibel, Charles A.
2590:J. Math. Soc. Jpn.
2564:Separable algebras
2451:, Theorem 4.2. If
2339:-subgroup algebra
2163:finitely generated
2125:
2012:strongly separable
1930:
1871:
1839:
1784:
1713:is separable over
1696:
1661:
1610:
1590:
1494:
1461:
1429:semisimple algebra
1374:finite-dimensional
1336:
1251:-monomorphisms of
1237:
1191:
1094:
1000:
918:
888:
821:
799:-algebra. For any
785:
723:
690:
638:
535:
471:
382:
359:
339:
315:
295:
269:
231:
177:
175:
81:
33:semisimple algebra
2685:978-0-521-55987-4
2614:10.1090/ulect/014
2573:978-1-4704-3770-1
2512:978-3-540-05371-2
2483:, Corollary 4.7.3
2267:that is split as
2144:symmetric algebra
2140:Frobenius algebra
1883:Maschke's theorem
1575:
1573:
1532:is invertible in
1377:division algebras
1352:Frobenius algebra
1257:algebraic closure
1247:are the distinct
1189:
1003:{\displaystyle a}
989:primitive element
930:elementary matrix
496:is equivalent to
385:{\displaystyle K}
362:{\displaystyle A}
342:{\displaystyle A}
318:{\displaystyle K}
272:{\displaystyle K}
29:separable algebra
2726:
2705:
2667:
2634:
2597:
2584:
2558:
2528:Samuel Eilenberg
2524:
2484:
2478:
2472:
2466:
2460:
2446:
2440:
2434:
2428:
2422:
2416:
2410:
2391:
2353:
2343:as the index is
2299:
2287:
2280:
2273:
2266:
2259:
2246:
2239:
2193:
2157:is commutative,
2134:
2132:
2131:
2126:
2124:
2123:
2111:
2110:
2100:
2095:
2077:
2076:
2064:
2063:
2053:
2048:
1989:
1962:
1951:
1939:
1937:
1936:
1931:
1908:
1907:
1880:
1878:
1877:
1872:
1848:
1846:
1845:
1840:
1835:
1834:
1793:
1791:
1790:
1785:
1774:
1773:
1705:
1703:
1702:
1697:
1695:
1694:
1670:
1668:
1667:
1662:
1660:
1659:
1619:
1617:
1616:
1611:
1609:
1608:
1589:
1574:
1572:
1555:
1503:
1501:
1500:
1495:
1490:
1489:
1470:
1468:
1467:
1464:{\textstyle L/K}
1462:
1457:
1345:
1343:
1342:
1337:
1326:
1325:
1315:
1310:
1246:
1244:
1243:
1238:
1236:
1235:
1217:
1216:
1200:
1198:
1197:
1192:
1190:
1188:
1178:
1169:
1168:
1159:
1157:
1156:
1147:
1146:
1136:
1125:
1103:
1101:
1100:
1095:
1093:
1092:
1083:
1082:
1072:
1061:
1009:
1007:
1006:
1001:
943:
927:
925:
924:
919:
917:
916:
897:
895:
894:
889:
887:
886:
871:
870:
857:
852:
830:
828:
827:
822:
794:
792:
791:
786:
775:
774:
732:
730:
729:
724:
716:
715:
699:
697:
696:
691:
689:
688:
687:
651:Such an element
647:
645:
644:
639:
631:
630:
618:
617:
602:
601:
589:
588:
544:
542:
541:
536:
528:
527:
518:
517:
492:is a section of
480:
478:
477:
472:
467:
466:
451:
450:
438:
437:
391:
389:
388:
383:
368:
366:
365:
360:
348:
346:
345:
340:
324:
322:
321:
316:
304:
302:
301:
296:
278:
276:
275:
270:
240:
238:
237:
232:
227:
226:
186:
184:
183:
178:
176:
149:
132:
131:
90:
88:
87:
82:
2734:
2733:
2729:
2728:
2727:
2725:
2724:
2723:
2709:
2708:
2686:
2670:
2657:
2637:
2624:
2600:
2587:
2574:
2561:
2537:
2513:
2503:Springer-Verlag
2496:
2493:
2488:
2487:
2479:
2475:
2467:
2463:
2447:
2443:
2439:, Theorem 4.4.1
2435:
2431:
2423:
2419:
2411:
2407:
2402:
2386:
2367:indecomposables
2348:
2289:
2282:
2275:
2268:
2261:
2251:
2241:
2231:
2228:
2226:Further results
2185:
2182:
2115:
2102:
2068:
2055:
2023:
2022:
2008:
1984:
1957:
1941:
1899:
1894:
1893:
1851:
1850:
1826:
1812:
1811:
1765:
1751:
1750:
1686:
1681:
1680:
1651:
1646:
1645:
1626:
1597:
1559:
1549:
1548:
1543:is a separable
1510:
1504:is semisimple.
1481:
1473:
1472:
1445:
1444:
1370:matrix algebras
1317:
1273:
1272:
1227:
1208:
1203:
1202:
1171:
1170:
1160:
1148:
1138:
1106:
1105:
1084:
1074:
1012:
1011:
992:
991:
971:if and only if
954:field extension
950:
933:
905:
900:
899:
875:
859:
833:
832:
801:
800:
795:is a separable
766:
761:
760:
747:non-commutative
739:
707:
702:
701:
675:
664:
663:
622:
609:
593:
580:
569:
568:
519:
509:
501:
500:
458:
442:
429:
400:
399:
374:
373:
351:
350:
331:
330:
307:
306:
281:
280:
261:
260:
218:
198:
197:
174:
173:
165:
160:
147:
146:
141:
136:
123:
118:
103:
102:
67:
66:
49:
21:
12:
11:
5:
2732:
2730:
2722:
2721:
2711:
2710:
2707:
2706:
2684:
2668:
2655:
2643:Maximal Orders
2635:
2622:
2598:
2585:
2572:
2559:
2535:
2525:
2511:
2492:
2489:
2486:
2485:
2473:
2461:
2441:
2429:
2417:
2404:
2403:
2401:
2398:
2274:, is split as
2227:
2224:
2220:formally étale
2210:(commutative)
2181:
2178:
2172:-module, then
2148:tensor algebra
2136:
2135:
2122:
2118:
2114:
2109:
2105:
2099:
2094:
2091:
2088:
2084:
2080:
2075:
2071:
2067:
2062:
2058:
2052:
2047:
2044:
2041:
2037:
2033:
2030:
2007:
2004:
1975:with elements
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1906:
1902:
1870:
1867:
1864:
1861:
1858:
1838:
1833:
1829:
1825:
1822:
1819:
1783:
1780:
1777:
1772:
1768:
1764:
1761:
1758:
1693:
1689:
1658:
1654:
1625:
1622:
1607:
1604:
1600:
1596:
1593:
1588:
1585:
1582:
1578:
1571:
1568:
1565:
1562:
1558:
1524:such that the
1509:
1506:
1493:
1488:
1484:
1480:
1460:
1456:
1452:
1397:characteristic
1335:
1332:
1329:
1324:
1320:
1314:
1309:
1306:
1303:
1299:
1295:
1292:
1289:
1286:
1283:
1280:
1271:is defined by
1234:
1230:
1226:
1223:
1220:
1215:
1211:
1187:
1184:
1181:
1177:
1174:
1167:
1163:
1155:
1151:
1145:
1141:
1135:
1132:
1129:
1124:
1121:
1118:
1114:
1091:
1087:
1081:
1077:
1071:
1068:
1065:
1060:
1057:
1054:
1050:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
999:
949:
946:
915:
912:
908:
885:
882:
878:
874:
869:
866:
862:
856:
851:
848:
845:
841:
820:
817:
814:
811:
808:
784:
781:
778:
773:
769:
738:
735:
722:
719:
714:
710:
686:
683:
678:
674:
671:
649:
648:
637:
634:
629:
625:
621:
616:
612:
608:
605:
600:
596:
592:
587:
583:
579:
576:
546:
545:
534:
531:
526:
522:
516:
512:
508:
482:
481:
470:
465:
461:
457:
454:
449:
445:
441:
436:
432:
428:
425:
422:
419:
416:
413:
410:
407:
381:
358:
338:
314:
294:
291:
288:
268:
242:
241:
230:
225:
221:
217:
214:
211:
208:
205:
188:
187:
172:
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166:
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161:
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156:
153:
150:
148:
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137:
135:
130:
126:
122:
119:
117:
114:
111:
110:
92:
91:
80:
77:
74:
48:
45:
13:
10:
9:
6:
4:
3:
2:
2731:
2720:
2717:
2716:
2714:
2703:
2699:
2695:
2691:
2687:
2681:
2677:
2673:
2669:
2666:
2662:
2658:
2656:0-19-852673-3
2652:
2648:
2644:
2640:
2636:
2633:
2629:
2625:
2623:0-8218-1962-3
2619:
2615:
2611:
2607:
2603:
2602:Kadison, Lars
2599:
2595:
2591:
2586:
2583:
2579:
2575:
2569:
2565:
2560:
2557:
2553:
2549:
2545:
2541:
2536:
2533:
2529:
2526:
2522:
2518:
2514:
2508:
2504:
2500:
2495:
2494:
2490:
2482:
2477:
2474:
2470:
2465:
2462:
2459:, Lemma 5.11.
2458:
2454:
2450:
2445:
2442:
2438:
2433:
2430:
2427:, p. 102
2426:
2421:
2418:
2414:
2409:
2406:
2399:
2397:
2395:
2389:
2384:
2380:
2376:
2372:
2368:
2364:
2360:
2356:
2351:
2346:
2342:
2338:
2334:
2330:
2327:-subgroup is
2326:
2322:
2318:
2313:
2311:
2307:
2303:
2297:
2293:
2288:are relative
2285:
2278:
2271:
2264:
2258:
2254:
2249:
2244:
2238:
2234:
2225:
2223:
2221:
2217:
2213:
2209:
2205:
2201:
2197:
2192:
2188:
2179:
2177:
2175:
2171:
2167:
2164:
2160:
2156:
2151:
2149:
2145:
2141:
2120:
2116:
2112:
2107:
2103:
2097:
2092:
2089:
2086:
2082:
2078:
2073:
2069:
2065:
2060:
2056:
2050:
2045:
2042:
2039:
2035:
2031:
2028:
2021:
2020:
2019:
2017:
2013:
2005:
2003:
2001:
1997:
1993:
1987:
1982:
1978:
1974:
1970:
1966:
1960:
1955:
1949:
1945:
1924:
1921:
1918:
1915:
1912:
1904:
1900:
1891:
1886:
1884:
1868:
1865:
1862:
1856:
1836:
1831:
1827:
1823:
1817:
1809:
1805:
1801:
1797:
1781:
1775:
1770:
1766:
1762:
1759:
1756:
1748:
1744:
1740:
1736:
1732:
1728:
1724:
1720:
1716:
1712:
1707:
1691:
1687:
1678:
1674:
1656:
1652:
1643:
1639:
1635:
1631:
1623:
1621:
1605:
1602:
1598:
1594:
1591:
1586:
1583:
1580:
1576:
1566:
1560:
1556:
1546:
1542:
1539:
1538:group algebra
1535:
1531:
1527:
1523:
1519:
1515:
1507:
1505:
1491:
1486:
1482:
1478:
1458:
1454:
1450:
1442:
1438:
1434:
1430:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1393:perfect field
1390:
1386:
1383:of the field
1382:
1378:
1375:
1371:
1367:
1364:
1359:
1357:
1353:
1349:
1330:
1322:
1318:
1312:
1307:
1304:
1301:
1297:
1293:
1287:
1281:
1278:
1270:
1266:
1262:
1258:
1254:
1250:
1232:
1228:
1224:
1221:
1218:
1213:
1209:
1182:
1175:
1172:
1165:
1161:
1153:
1149:
1143:
1139:
1133:
1130:
1127:
1122:
1119:
1116:
1112:
1089:
1085:
1079:
1075:
1069:
1066:
1063:
1058:
1055:
1052:
1048:
1041:
1038:
1035:
1029:
1023:
1017:
997:
990:
986:
982:
979:-algebra. If
978:
974:
970:
966:
962:
958:
955:
947:
945:
941:
937:
931:
913:
910:
906:
883:
880:
876:
872:
867:
864:
860:
854:
849:
846:
843:
839:
818:
815:
812:
809:
806:
798:
779:
771:
767:
759:
756:
752:
748:
744:
736:
734:
720:
717:
712:
708:
700:it satisfies
676:
672:
669:
661:
660:
657:separability
654:
635:
632:
627:
623:
619:
614:
610:
606:
603:
598:
594:
590:
585:
581:
577:
574:
567:
566:
565:
563:
559:
555:
551:
532:
529:
524:
520:
514:
510:
506:
499:
498:
497:
495:
491:
487:
468:
463:
459:
455:
452:
447:
443:
439:
434:
430:
426:
423:
417:
411:
408:
405:
398:
397:
396:
393:
379:
372:
356:
336:
328:
312:
292:
286:
266:
257:
255:
251:
247:
228:
223:
219:
215:
209:
206:
203:
196:
195:
194:
193:
170:
167:
157:
154:
151:
143:
133:
128:
124:
120:
115:
112:
101:
100:
99:
97:
78:
72:
65:
64:
63:
62:
58:
54:
46:
44:
42:
38:
34:
31:is a kind of
30:
26:
19:
18:étale algebra
2675:
2642:
2605:
2593:
2589:
2563:
2547:
2543:
2498:
2476:
2464:
2457:Kadison 1999
2452:
2444:
2432:
2420:
2408:
2393:
2387:
2382:
2378:
2374:
2370:
2362:
2354:
2349:
2340:
2336:
2332:
2324:
2320:
2316:
2314:
2309:
2305:
2301:
2295:
2291:
2283:
2276:
2269:
2262:
2256:
2252:
2247:
2242:
2236:
2232:
2229:
2215:
2211:
2203:
2199:
2190:
2186:
2183:
2173:
2169:
2158:
2154:
2152:
2137:
2015:
2011:
2009:
1999:
1995:
1991:
1985:
1980:
1976:
1972:
1968:
1964:
1958:
1956:is zero for
1953:
1947:
1943:
1887:
1807:
1803:
1799:
1795:
1746:
1742:
1738:
1734:
1726:
1722:
1714:
1710:
1708:
1672:
1633:
1629:
1627:
1544:
1540:
1533:
1529:
1522:finite group
1517:
1513:
1511:
1471:the algebra
1440:
1436:
1432:
1424:
1420:
1416:
1412:
1408:
1401:finite field
1388:
1384:
1365:
1360:
1355:
1347:
1268:
1264:
1260:
1252:
1248:
984:
980:
976:
972:
960:
956:
951:
939:
935:
928:denotes the
796:
754:
750:
742:
740:
656:
655:is called a
652:
650:
561:
557:
553:
549:
547:
493:
489:
485:
483:
394:
370:
259:If the ring
258:
249:
245:
243:
189:
95:
93:
53:homomorphism
50:
28:
22:
2550:: 233–242,
2425:Reiner 2003
1983:satisfying
1536:, then the
1508:Group rings
1399:zero, or a
57:commutative
25:mathematics
2665:1024.16008
2639:Reiner, I.
2521:0215.36602
2491:References
2359:isomorphic
2168:separable
2166:projective
2018:, meaning
1638:projective
963:of finite
749:) ring of
659:idempotent
349:, we call
94:is called
2596:: 360–373
2481:Ford 2017
2469:Ford 2017
2437:Ford 2017
2413:Ford 2017
2400:Citations
2390:-bimodule
2214:-algebra
2113:⊗
2083:∑
2066:⊗
2036:∑
2016:symmetric
1866:⊗
1860:↦
1849:given by
1828:⊗
1821:→
1779:→
1767:⊗
1757:μ
1632:-algebra
1603:−
1595:⊗
1584:∈
1577:∑
1483:⊗
1439:-algebra
1319:σ
1298:∑
1229:σ
1222:…
1210:σ
1150:⊗
1131:−
1113:∑
1067:−
1049:∑
1039:−
873:⊗
840:∑
816:≤
810:≤
673:⊗
620:⊗
607:∑
591:⊗
575:∑
507:∑
460:⊗
453:∈
440:⊗
427:∑
412:σ
325:into the
290:→
254:bimodules
220:⊗
213:→
204:σ
190:admits a
163:↦
155:⊗
139:→
125:⊗
113:μ
96:separable
76:→
2719:Algebras
2713:Category
2702:36131259
2674:(1994).
2641:(2003),
2604:(1999),
2286:-modules
2279:-modules
2272:-modules
2265:-modules
1403:, or an
1255:into an
1176:′
898:, where
758:matrices
737:Examples
2694:1269324
2632:1690111
2582:3618889
2556:0227211
2352:-module
2345:coprime
2245:-module
1892:groups
745:, the (
192:section
2700:
2692:
2682:
2663:
2653:
2630:
2620:
2580:
2570:
2554:
2519:
2509:
2415:, §4.2
2329:cyclic
1990:, but
1961:> 0
1642:module
987:has a
965:degree
494:μ
490:σ
486:σ
327:center
2385:is a
2161:is a
1731:split
1526:order
1520:is a
1431:over
1391:is a
1372:over
1363:field
1354:over
1350:as a
1267:into
967:is a
305:maps
61:rings
2698:OCLC
2680:ISBN
2651:ISBN
2618:ISBN
2568:ISBN
2507:ISBN
2308:has
2208:flat
1979:and
1677:flat
753:-by-
256:.
27:, a
2661:Zbl
2610:doi
2517:Zbl
2357:is
2218:is
2153:If
2150:).
2000:bRa
1988:= 1
1940:of
1733:as
1721:of
1644:of
1528:of
1512:If
1259:of
560:in
329:of
23:In
2715::
2696:.
2690:MR
2688:.
2659:,
2649:,
2628:MR
2626:,
2616:,
2594:18
2592:,
2578:MR
2576:,
2552:MR
2546:,
2542:,
2515:.
2505:.
2396:.
2294:,
2255:|
2235:|
2222:.
2189:/
2002:.
1992:ba
1986:ab
1946:,
1620:.
1358:.
952:A
938:,
733:.
564::
409::=
392:.
369:a
59:)
51:A
43:.
2704:.
2612::
2548:4
2523:.
2453:A
2394:A
2388:B
2383:B
2379:B
2375:A
2371:M
2363:B
2355:M
2350:A
2341:B
2337:p
2333:p
2325:p
2321:p
2317:A
2310:R
2306:S
2302:R
2298:)
2296:S
2292:R
2290:(
2284:R
2277:R
2270:S
2263:R
2257:S
2253:R
2248:R
2243:S
2237:S
2233:R
2216:A
2212:K
2204:K
2200:A
2191:K
2187:A
2174:A
2170:K
2159:A
2155:K
2121:i
2117:x
2108:i
2104:y
2098:n
2093:1
2090:=
2087:i
2079:=
2074:i
2070:y
2061:i
2057:x
2051:n
2046:1
2043:=
2040:i
2032:=
2029:e
1996:S
1981:b
1977:a
1973:R
1969:S
1965:R
1959:n
1954:M
1950:)
1948:S
1944:R
1942:(
1928:)
1925:M
1922:;
1919:S
1916:,
1913:R
1910:(
1905:n
1901:H
1869:1
1863:a
1857:a
1837:A
1832:K
1824:A
1818:A
1808:K
1806:-
1804:A
1800:A
1798:-
1796:A
1782:A
1776:A
1771:K
1763:A
1760::
1747:A
1745:-
1743:A
1739:K
1737:-
1735:A
1727:A
1725:-
1723:A
1715:K
1711:A
1692:e
1688:A
1673:A
1657:e
1653:A
1634:A
1630:K
1606:1
1599:g
1592:g
1587:G
1581:g
1570:)
1567:G
1564:(
1561:o
1557:1
1545:K
1541:K
1534:K
1530:G
1518:G
1514:K
1492:L
1487:K
1479:A
1459:K
1455:/
1451:L
1441:A
1437:K
1433:K
1425:K
1421:K
1417:K
1413:K
1409:K
1389:K
1385:K
1366:K
1356:K
1348:L
1334:)
1331:x
1328:(
1323:i
1313:n
1308:1
1305:=
1302:i
1294:=
1291:)
1288:x
1285:(
1282:r
1279:T
1269:K
1265:L
1261:K
1253:L
1249:K
1233:n
1225:,
1219:,
1214:1
1186:)
1183:a
1180:(
1173:p
1166:i
1162:b
1154:K
1144:i
1140:a
1134:1
1128:n
1123:0
1120:=
1117:i
1090:i
1086:x
1080:i
1076:b
1070:1
1064:n
1059:0
1056:=
1053:i
1045:)
1042:a
1036:x
1033:(
1030:=
1027:)
1024:x
1021:(
1018:p
998:a
985:K
983:/
981:L
977:K
973:L
961:K
959:/
957:L
942:)
940:j
936:i
934:(
914:j
911:i
907:e
884:i
881:j
877:e
868:j
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861:e
855:n
850:1
847:=
844:i
819:n
813:j
807:1
797:R
783:)
780:R
777:(
772:n
768:M
755:n
751:n
743:R
721:p
718:=
713:2
709:p
685:p
682:o
677:A
670:A
653:p
636:.
633:a
628:i
624:b
615:i
611:a
604:=
599:i
595:b
586:i
582:a
578:a
562:A
558:a
554:A
552:-
550:A
533:1
530:=
525:i
521:b
515:i
511:a
469:A
464:K
456:A
448:i
444:b
435:i
431:a
424:=
421:)
418:1
415:(
406:p
380:K
357:A
337:A
313:K
293:A
287:K
267:K
252:-
250:A
248:-
246:A
229:A
224:K
216:A
210:A
207::
171:b
168:a
158:b
152:a
144:A
134:A
129:K
121:A
116::
79:A
73:K
20:.
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