963:
757:
1229:
2113:
549:
827:
1532:
838:
1282:
1118:
642:
1352:
1010:
1811:
1642:
238:
1602:
1126:
1559:
1068:
1037:
630:
579:
440:
2087:
1851:
1831:
1773:
1753:
1733:
1713:
1689:
1662:
1462:
1407:
1302:
599:
468:
413:
393:
2008:
476:
765:
2029:
2152:
1370:
Since a finite-dimensional algebra over a field is
Artinian, the Wedderburn–Artin theorem implies that every finite-dimensional
1471:
2059:
958:{\displaystyle \mathrm {End} {\big (}I_{i}^{\oplus n_{i}}{\big )}\;\cong \;M_{n_{i}}{\big (}\mathrm {End} (I_{i}){\big )}}
752:{\displaystyle \mathrm {End} (R_{R})\;\cong \;\bigoplus _{i=1}^{m}\mathrm {End} {\big (}I_{i}^{\oplus n_{i}}{\big )}}
1433:
1237:
1073:
1918:. For Artinian rings, the two notions are equivalent, so "Artinian" is included here to eliminate that ambiguity.
1355:
1872:
1311:
971:
1668:
326:
39:
1778:
1607:
193:
1862:
830:
307:
1877:
447:
273:
1564:
2046:
369:
47:
2000:
1667:
Furthermore, the
Wedderburn–Artin theorem reduces the problem of classifying finite-dimensional
1224:{\displaystyle R\;\cong \;\bigoplus _{i=1}^{m}M_{n_{i}}{\big (}\mathrm {End} (I_{i}){\big )}\,.}
2078:
2004:
1887:
1418:
1436:
is the field itself, the
Wedderburn–Artin theorem has strong consequences in this case. Let
2130:
2122:
2096:
2038:
1915:
1305:
1040:
299:
55:
1537:
1046:
1015:
608:
557:
418:
2134:
1914:. However, some authors use "semisimple" differently, to mean that the ring has a trivial
1907:
1441:
159:
43:
1993:
1836:
1816:
1758:
1738:
1718:
1698:
1674:
1647:
1447:
1392:
1371:
1287:
584:
453:
398:
378:
353:
2146:
1911:
1426:
1359:
443:
372:
of the
Wedderburn–Artin theorem. A common modern one takes the following approach.
241:
132:
117:
84:
1354:, but the proof would still go through. To see this proof in a larger context, see
1867:
633:
17:
1382:
357:
188:
128:
113:
80:
2108:
1882:
1422:
2100:
272:
There is also a version of the
Wedderburn–Artin theorem for algebras over a
1444:
that is a finite-dimensional algebra over an algebraically closed field
2126:
2050:
544:{\displaystyle R_{R}\;\cong \;\bigoplus _{i=1}^{m}I_{i}^{\oplus n_{i}}}
265:, both of which are uniquely determined up to permutation of the index
108:, both of which are uniquely determined up to permutation of the index
31:
1755:. It implies that any finite-dimensional central simple algebra over
2082:
2042:
2114:
1425:
later generalized this result to the case of simple left or right
822:{\displaystyle \mathrm {End} {\big (}I_{i}^{\oplus n_{i}}{\big )}}
2027:
Henderson, D.W. (1965). "A short proof of
Wedderburn's theorem".
340:
is a finite-dimensional simple algebra over a division ring
1432:
Since the only finite-dimensional division algebra over an
50:. The theorem states that an (Artinian) semisimple ring
1527:{\displaystyle \textstyle \prod _{i=1}^{r}M_{n_{i}}(k)}
1833:
is a finite-dimensional central division algebra over
1782:
1475:
1839:
1819:
1781:
1761:
1741:
1721:
1701:
1677:
1650:
1610:
1567:
1540:
1474:
1450:
1395:
1314:
1290:
1240:
1129:
1076:
1049:
1018:
974:
841:
768:
645:
611:
587:
560:
479:
456:
421:
401:
381:
196:
1937:
1935:
1358:. For the proof of an important special case, see
1992:
1845:
1825:
1805:
1767:
1747:
1727:
1707:
1683:
1656:
1636:
1596:
1553:
1526:
1456:
1401:
1346:
1296:
1276:
1223:
1112:
1062:
1031:
1004:
957:
821:
751:
624:
593:
573:
543:
462:
434:
407:
387:
232:
2111:(1927). "Zur Theorie der hyperkomplexen Zahlen".
1691:to the problem of classifying finite-dimensional
356:are finite-dimensional simple algebras over the
162:. Then the Wedderburn–Artin theorem states that
27:Classification of semi-simple rings and algebras
2060:"A short proof of the Wedderburn–Artin theorem"
298:in the above statement is a finite-dimensional
2088:Proceedings of the London Mathematical Society
1385:over some finite-dimensional division algebra
1212:
1178:
950:
916:
887:
855:
814:
782:
744:
712:
8:
1277:{\displaystyle R\cong \mathrm {End} (R_{R})}
1113:{\displaystyle R\cong \mathrm {End} (R_{R})}
166:is isomorphic to a product of finitely many
1417:are uniquely determined. This was shown by
1995:Introductory Lectures on Rings and Modules
1137:
1133:
896:
892:
677:
673:
494:
490:
1963:
1952:
1838:
1818:
1787:
1780:
1760:
1740:
1720:
1700:
1676:
1649:
1628:
1615:
1609:
1577:
1572:
1566:
1545:
1539:
1506:
1501:
1491:
1480:
1473:
1449:
1394:
1332:
1330:
1315:
1313:
1289:
1265:
1247:
1239:
1217:
1211:
1210:
1201:
1183:
1177:
1176:
1168:
1163:
1153:
1142:
1128:
1101:
1083:
1075:
1054:
1048:
1023:
1017:
993:
975:
973:
949:
948:
939:
921:
915:
914:
906:
901:
886:
885:
877:
869:
864:
854:
853:
842:
840:
813:
812:
804:
796:
791:
781:
780:
769:
767:
743:
742:
734:
726:
721:
711:
710:
699:
693:
682:
664:
646:
644:
616:
610:
586:
565:
559:
533:
525:
520:
510:
499:
484:
478:
455:
426:
420:
400:
380:
221:
206:
201:
195:
2020:Basic Algebra: Groups, Rings, and Fields
581:are mutually nonisomorphic simple right
442:is isomorphic to a finite direct sum of
1931:
1899:
1347:{\displaystyle \mathrm {End} ({}_{R}R)}
1999:. Cambridge University Press. p.
1941:
352:. For example, matrix rings over the
1005:{\displaystyle \mathrm {End} (I_{i})}
446:(which are the same as minimal right
7:
1974:
1806:{\displaystyle \textstyle M_{n}(D)}
1234:Here we used right modules because
605:th one appearing with multiplicity
283:is a finite-dimensional semisimple
1775:is isomorphic to a matrix algebra
1715:: that is, division algebras over
1322:
1319:
1316:
1254:
1251:
1248:
1190:
1187:
1184:
1090:
1087:
1084:
982:
979:
976:
928:
925:
922:
849:
846:
843:
776:
773:
770:
706:
703:
700:
653:
650:
647:
25:
2030:The American Mathematical Monthly
1637:{\displaystyle n_{i}\times n_{i}}
1374:over a field is isomorphic to an
632:. This gives an isomorphism of
233:{\displaystyle M_{n_{i}}(D_{i})}
2058:Nicholson, William K. (1993).
1799:
1793:
1591:
1585:
1520:
1514:
1341:
1326:
1271:
1258:
1207:
1194:
1107:
1094:
999:
986:
945:
932:
670:
657:
395:is semisimple. Then the right
227:
214:
1:
1906:By the definition used here,
470:). Write this direct sum as
1597:{\displaystyle M_{n_{i}}(k)}
968:where the endomorphism ring
1304:would be isomorphic to the
2169:
1561:are positive integers and
1434:algebraically closed field
1284:; if we used left modules
2083:"On Hypercomplex Numbers"
1356:Decomposition of a module
348:need not be contained in
146:are uniquely determined.
1991:Beachy, John A. (1999).
1873:Jacobson density theorem
36:Wedderburn–Artin theorem
2153:Theorems in ring theory
1695:division algebras over
1669:central simple algebras
2101:10.1112/plms/s2-6.1.77
1847:
1827:
1807:
1769:
1749:
1729:
1709:
1685:
1658:
1638:
1598:
1555:
1528:
1496:
1458:
1403:
1348:
1298:
1278:
1225:
1158:
1114:
1064:
1039:is a division ring by
1033:
1006:
959:
823:
753:
698:
626:
595:
575:
545:
515:
464:
436:
409:
389:
234:
112:. In particular, any
40:classification theorem
1848:
1828:
1808:
1770:
1750:
1730:
1710:
1686:
1659:
1639:
1599:
1556:
1554:{\displaystyle n_{i}}
1529:
1476:
1459:
1404:
1349:
1299:
1279:
1226:
1138:
1115:
1065:
1063:{\displaystyle I_{i}}
1034:
1032:{\displaystyle I_{i}}
1007:
960:
824:
754:
678:
627:
625:{\displaystyle n_{i}}
596:
576:
574:{\displaystyle I_{i}}
546:
495:
465:
437:
435:{\displaystyle R_{R}}
410:
390:
235:
2018:Cohn, P. M. (2003).
1837:
1817:
1779:
1759:
1739:
1719:
1699:
1675:
1648:
1608:
1565:
1538:
1472:
1468:is a finite product
1448:
1393:
1360:Simple Artinian ring
1312:
1288:
1238:
1127:
1074:
1047:
1016:
972:
839:
766:
762:and we can identify
643:
609:
585:
558:
477:
454:
419:
399:
379:
287:-algebra, then each
254:, for some integers
194:
120:is isomorphic to an
97:, for some integers
2067:New Zealand J. Math
2022:. pp. 137–139.
1878:Hypercomplex number
1070:is simple. Since
884:
811:
741:
540:
54:is isomorphic to a
48:semisimple algebras
18:Semisimple Artinian
2127:10.1007/BF02952526
2079:Wedderburn, J.H.M.
1910:are automatically
1843:
1823:
1803:
1802:
1765:
1745:
1725:
1705:
1681:
1654:
1634:
1604:is the algebra of
1594:
1551:
1524:
1523:
1454:
1399:
1344:
1294:
1274:
1221:
1110:
1060:
1029:
1002:
955:
860:
819:
787:
749:
717:
622:
591:
571:
541:
516:
460:
432:
405:
385:
368:There are various
230:
2010:978-0-521-64407-5
1888:Joseph Wedderburn
1863:Maschke's theorem
1846:{\displaystyle k}
1826:{\displaystyle D}
1768:{\displaystyle k}
1748:{\displaystyle k}
1728:{\displaystyle k}
1708:{\displaystyle k}
1684:{\displaystyle k}
1657:{\displaystyle k}
1457:{\displaystyle k}
1419:Joseph Wedderburn
1402:{\displaystyle k}
1297:{\displaystyle R}
594:{\displaystyle R}
463:{\displaystyle R}
408:{\displaystyle R}
388:{\displaystyle R}
375:Suppose the ring
58:of finitely many
16:(Redirected from
2160:
2138:
2104:
2074:
2064:
2054:
2023:
2014:
1998:
1977:
1972:
1966:
1961:
1955:
1950:
1944:
1939:
1919:
1916:Jacobson radical
1908:semisimple rings
1904:
1852:
1850:
1849:
1844:
1832:
1830:
1829:
1824:
1812:
1810:
1809:
1804:
1792:
1791:
1774:
1772:
1771:
1766:
1754:
1752:
1751:
1746:
1735:whose center is
1734:
1732:
1731:
1726:
1714:
1712:
1711:
1706:
1690:
1688:
1687:
1682:
1663:
1661:
1660:
1655:
1643:
1641:
1640:
1635:
1633:
1632:
1620:
1619:
1603:
1601:
1600:
1595:
1584:
1583:
1582:
1581:
1560:
1558:
1557:
1552:
1550:
1549:
1533:
1531:
1530:
1525:
1513:
1512:
1511:
1510:
1495:
1490:
1467:
1463:
1461:
1460:
1455:
1439:
1408:
1406:
1405:
1400:
1353:
1351:
1350:
1345:
1337:
1336:
1331:
1325:
1306:opposite algebra
1303:
1301:
1300:
1295:
1283:
1281:
1280:
1275:
1270:
1269:
1257:
1230:
1228:
1227:
1222:
1216:
1215:
1206:
1205:
1193:
1182:
1181:
1175:
1174:
1173:
1172:
1157:
1152:
1119:
1117:
1116:
1111:
1106:
1105:
1093:
1069:
1067:
1066:
1061:
1059:
1058:
1038:
1036:
1035:
1030:
1028:
1027:
1011:
1009:
1008:
1003:
998:
997:
985:
964:
962:
961:
956:
954:
953:
944:
943:
931:
920:
919:
913:
912:
911:
910:
891:
890:
883:
882:
881:
868:
859:
858:
852:
828:
826:
825:
820:
818:
817:
810:
809:
808:
795:
786:
785:
779:
758:
756:
755:
750:
748:
747:
740:
739:
738:
725:
716:
715:
709:
697:
692:
669:
668:
656:
631:
629:
628:
623:
621:
620:
604:
600:
598:
597:
592:
580:
578:
577:
572:
570:
569:
550:
548:
547:
542:
539:
538:
537:
524:
514:
509:
489:
488:
469:
467:
466:
461:
441:
439:
438:
433:
431:
430:
414:
412:
411:
406:
394:
392:
391:
386:
339:
332:
327:finite extension
325:; it could be a
324:
320:
305:
300:division algebra
297:
286:
282:
278:
268:
264:
253:
239:
237:
236:
231:
226:
225:
213:
212:
211:
210:
187:
176:
165:
158:be a (Artinian)
157:
111:
107:
96:
79:
68:
44:semisimple rings
21:
2168:
2167:
2163:
2162:
2161:
2159:
2158:
2157:
2143:
2142:
2141:
2107:
2077:
2062:
2057:
2043:10.2307/2313499
2026:
2017:
2011:
1990:
1986:
1981:
1980:
1973:
1969:
1962:
1958:
1951:
1947:
1940:
1933:
1928:
1923:
1922:
1905:
1901:
1896:
1859:
1835:
1834:
1815:
1814:
1783:
1777:
1776:
1757:
1756:
1737:
1736:
1717:
1716:
1697:
1696:
1673:
1672:
1646:
1645:
1624:
1611:
1606:
1605:
1573:
1568:
1563:
1562:
1541:
1536:
1535:
1502:
1497:
1470:
1469:
1465:
1446:
1445:
1442:semisimple ring
1437:
1391:
1390:
1368:
1329:
1310:
1309:
1286:
1285:
1261:
1236:
1235:
1197:
1164:
1159:
1125:
1124:
1097:
1072:
1071:
1050:
1045:
1044:
1019:
1014:
1013:
989:
970:
969:
935:
902:
897:
873:
837:
836:
829:with a ring of
800:
764:
763:
730:
660:
641:
640:
612:
607:
606:
602:
583:
582:
561:
556:
555:
529:
480:
475:
474:
452:
451:
422:
417:
416:
397:
396:
377:
376:
366:
354:complex numbers
337:
330:
322:
319:
311:
303:
296:
288:
284:
280:
276:
266:
263:
255:
252:
244:
217:
202:
197:
192:
191:
186:
178:
175:
167:
163:
160:semisimple ring
155:
152:
109:
106:
98:
95:
87:
78:
70:
67:
59:
28:
23:
22:
15:
12:
11:
5:
2166:
2164:
2156:
2155:
2145:
2144:
2140:
2139:
2105:
2075:
2055:
2037:(4): 385–386.
2024:
2015:
2009:
1987:
1985:
1982:
1979:
1978:
1967:
1964:Nicholson 1993
1956:
1953:Henderson 1965
1945:
1930:
1929:
1927:
1924:
1921:
1920:
1912:Artinian rings
1898:
1897:
1895:
1892:
1891:
1890:
1885:
1880:
1875:
1870:
1865:
1858:
1855:
1842:
1822:
1801:
1798:
1795:
1790:
1786:
1764:
1744:
1724:
1704:
1680:
1653:
1644:matrices over
1631:
1627:
1623:
1618:
1614:
1593:
1590:
1587:
1580:
1576:
1571:
1548:
1544:
1522:
1519:
1516:
1509:
1505:
1500:
1494:
1489:
1486:
1483:
1479:
1453:
1427:Artinian rings
1398:
1372:simple algebra
1367:
1364:
1343:
1340:
1335:
1328:
1324:
1321:
1318:
1293:
1273:
1268:
1264:
1260:
1256:
1253:
1250:
1246:
1243:
1232:
1231:
1220:
1214:
1209:
1204:
1200:
1196:
1192:
1189:
1186:
1180:
1171:
1167:
1162:
1156:
1151:
1148:
1145:
1141:
1136:
1132:
1109:
1104:
1100:
1096:
1092:
1089:
1086:
1082:
1079:
1057:
1053:
1026:
1022:
1001:
996:
992:
988:
984:
981:
978:
966:
965:
952:
947:
942:
938:
934:
930:
927:
924:
918:
909:
905:
900:
895:
889:
880:
876:
872:
867:
863:
857:
851:
848:
845:
816:
807:
803:
799:
794:
790:
784:
778:
775:
772:
760:
759:
746:
737:
733:
729:
724:
720:
714:
708:
705:
702:
696:
691:
688:
685:
681:
676:
672:
667:
663:
659:
655:
652:
649:
619:
615:
601:-modules, the
590:
568:
564:
552:
551:
536:
532:
528:
523:
519:
513:
508:
505:
502:
498:
493:
487:
483:
459:
444:simple modules
429:
425:
404:
384:
365:
362:
315:
292:
259:
248:
242:division rings
229:
224:
220:
216:
209:
205:
200:
182:
171:
151:
148:
116:left or right
102:
91:
85:division rings
74:
63:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2165:
2154:
2151:
2150:
2148:
2136:
2132:
2128:
2124:
2120:
2116:
2115:
2110:
2106:
2102:
2098:
2094:
2090:
2089:
2084:
2080:
2076:
2072:
2068:
2061:
2056:
2052:
2048:
2044:
2040:
2036:
2032:
2031:
2025:
2021:
2016:
2012:
2006:
2002:
1997:
1996:
1989:
1988:
1983:
1976:
1971:
1968:
1965:
1960:
1957:
1954:
1949:
1946:
1943:
1938:
1936:
1932:
1925:
1917:
1913:
1909:
1903:
1900:
1893:
1889:
1886:
1884:
1881:
1879:
1876:
1874:
1871:
1869:
1866:
1864:
1861:
1860:
1856:
1854:
1840:
1820:
1796:
1788:
1784:
1762:
1742:
1722:
1702:
1694:
1678:
1671:over a field
1670:
1665:
1651:
1629:
1625:
1621:
1616:
1612:
1588:
1578:
1574:
1569:
1546:
1542:
1517:
1507:
1503:
1498:
1492:
1487:
1484:
1481:
1477:
1451:
1443:
1435:
1430:
1428:
1424:
1420:
1416:
1412:
1409:, where both
1396:
1388:
1384:
1381:
1377:
1373:
1365:
1363:
1361:
1357:
1338:
1333:
1307:
1291:
1266:
1262:
1244:
1241:
1218:
1202:
1198:
1169:
1165:
1160:
1154:
1149:
1146:
1143:
1139:
1134:
1130:
1123:
1122:
1121:
1102:
1098:
1080:
1077:
1055:
1051:
1042:
1041:Schur's lemma
1024:
1020:
994:
990:
940:
936:
907:
903:
898:
893:
878:
874:
870:
865:
861:
835:
834:
833:
832:
805:
801:
797:
792:
788:
735:
731:
727:
722:
718:
694:
689:
686:
683:
679:
674:
665:
661:
639:
638:
637:
635:
617:
613:
588:
566:
562:
534:
530:
526:
521:
517:
511:
506:
503:
500:
496:
491:
485:
481:
473:
472:
471:
457:
449:
445:
427:
423:
402:
382:
373:
371:
363:
361:
359:
355:
351:
347:
343:
336:Note that if
334:
328:
318:
314:
309:
301:
295:
291:
275:
270:
262:
258:
251:
247:
243:
222:
218:
207:
203:
198:
190:
185:
181:
174:
170:
161:
149:
147:
145:
141:
138:, where both
137:
134:
133:division ring
130:
127:
123:
119:
118:Artinian ring
115:
105:
101:
94:
90:
86:
82:
77:
73:
66:
62:
57:
53:
49:
45:
41:
37:
33:
19:
2118:
2112:
2092:
2086:
2070:
2066:
2034:
2028:
2019:
1994:
1970:
1959:
1948:
1902:
1868:Brauer group
1692:
1666:
1431:
1414:
1410:
1386:
1379:
1375:
1369:
1366:Consequences
1233:
1120:we conclude
967:
761:
634:endomorphism
553:
374:
367:
358:real numbers
349:
345:
341:
335:
321:need not be
316:
312:
293:
289:
271:
260:
256:
249:
245:
189:matrix rings
183:
179:
172:
168:
153:
143:
139:
135:
125:
121:
103:
99:
92:
88:
81:matrix rings
75:
71:
64:
60:
51:
35:
29:
2121:: 251–260.
1942:Beachy 1999
1383:matrix ring
129:matrix ring
2135:53.0114.03
2095:: 77–118.
1984:References
1883:Emil Artin
1534:where the
1423:Emil Artin
1043:, because
554:where the
2109:Artin, E.
1975:Cohn 2003
1926:Citations
1622:×
1478:∏
1245:≅
1140:⨁
1135:≅
1081:≅
894:≅
871:⊕
798:⊕
728:⊕
680:⨁
675:≅
527:⊕
497:⨁
492:≅
2147:Category
2081:(1908).
2073:: 83–86.
1857:See also
1464:. Then
831:matrices
415:-module
310:of each
2051:2313499
1693:central
306:. The
150:Theorem
131:over a
56:product
32:algebra
2133:
2049:
2007:
1813:where
636:rings
448:ideals
370:proofs
308:center
114:simple
34:, the
2063:(PDF)
2047:JSTOR
1894:Notes
1664:.
1440:be a
1389:over
364:Proof
302:over
279:. If
274:field
240:over
83:over
38:is a
2005:ISBN
1413:and
1378:-by-
177:-by-
154:Let
142:and
124:-by-
69:-by-
46:and
42:for
2131:JFM
2123:doi
2097:doi
2039:doi
2001:156
1429:.
1308:of
1012:of
450:of
329:of
269:.
30:In
2149::
2129:.
2117:.
2091:.
2085:.
2071:22
2069:.
2065:.
2045:.
2035:72
2033:.
2003:.
1934:^
1853:.
1421:.
1362:.
360:.
344:,
333:.
2137:.
2125::
2119:5
2103:.
2099::
2093:6
2053:.
2041::
2013:.
1841:k
1821:D
1800:)
1797:D
1794:(
1789:n
1785:M
1763:k
1743:k
1723:k
1703:k
1679:k
1652:k
1630:i
1626:n
1617:i
1613:n
1592:)
1589:k
1586:(
1579:i
1575:n
1570:M
1547:i
1543:n
1521:)
1518:k
1515:(
1508:i
1504:n
1499:M
1493:r
1488:1
1485:=
1482:i
1466:R
1452:k
1438:R
1415:D
1411:n
1397:k
1387:D
1380:n
1376:n
1342:)
1339:R
1334:R
1327:(
1323:d
1320:n
1317:E
1292:R
1272:)
1267:R
1263:R
1259:(
1255:d
1252:n
1249:E
1242:R
1219:.
1213:)
1208:)
1203:i
1199:I
1195:(
1191:d
1188:n
1185:E
1179:(
1170:i
1166:n
1161:M
1155:m
1150:1
1147:=
1144:i
1131:R
1108:)
1103:R
1099:R
1095:(
1091:d
1088:n
1085:E
1078:R
1056:i
1052:I
1025:i
1021:I
1000:)
995:i
991:I
987:(
983:d
980:n
977:E
951:)
946:)
941:i
937:I
933:(
929:d
926:n
923:E
917:(
908:i
904:n
899:M
888:)
879:i
875:n
866:i
862:I
856:(
850:d
847:n
844:E
815:)
806:i
802:n
793:i
789:I
783:(
777:d
774:n
771:E
745:)
736:i
732:n
723:i
719:I
713:(
707:d
704:n
701:E
695:m
690:1
687:=
684:i
671:)
666:R
662:R
658:(
654:d
651:n
648:E
618:i
614:n
603:i
589:R
567:i
563:I
535:i
531:n
522:i
518:I
512:m
507:1
504:=
501:i
486:R
482:R
458:R
428:R
424:R
403:R
383:R
350:E
346:D
342:E
338:R
331:k
323:k
317:i
313:D
304:k
294:i
290:D
285:k
281:R
277:k
267:i
261:i
257:n
250:i
246:D
228:)
223:i
219:D
215:(
208:i
204:n
199:M
184:i
180:n
173:i
169:n
164:R
156:R
144:D
140:n
136:D
126:n
122:n
110:i
104:i
100:n
93:i
89:D
76:i
72:n
65:i
61:n
52:R
20:)
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