1439:
1227:
1296:
1146:
184:
39:, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules.
996:
modules. Using these facts, many definitions and statements about modules can be immediately translated into definitions and statements about bimodules. For example, the
1444:
holds whenever either (and hence the other) side of the equation is defined, and where ∘ is the usual composition of homomorphisms. In this interpretation, the category
1338:
981:(where the multiplication is defined with the arguments exchanged). Bimodule homomorphisms are the same as homomorphisms of left
1091:), and one can hence construct a category whose objects are the rings and whose morphisms are the bimodules. This is in fact a
563:
is a left module, we can define multiplication on the right to be the same as multiplication on the left. (However, not all
511:
by taking the left and right actions to be multiplication – the actions commute by associativity. This can be extended to
124:
1770:
35:, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of
1157:
1233:
328:; the heights and widths of the matrices have been chosen so that multiplication is defined. Note that
1119:
1568:
1023:
1589:
1789:
997:
1022:
There are however some new effects in the world of bimodules, especially when it comes to the
325:
1016:
403:, is the statement that multiplication of matrices is associative (which, in the case of a
248:
32:
8:
1543:
531:
1739:
1738:
Street, Ross (20 Mar 2003). "Categorical and combinatorial aspects of descent theory".
56:
598:
1766:
1642:
1468:
1495:
1012:
553:
520:
20:
1758:
1686:
321:
1332:. One immediately verifies the interchange law for bimodule homomorphisms, i.e.
549:, with the ring multiplication both as the left and as the right multiplication.
1483:
1434:{\displaystyle (f'\otimes g')\circ (f\otimes g)=(f'\circ f)\otimes (g'\circ g)}
320:
matrices. Addition and multiplication are carried out using the usual rules of
660:
is a ring with the multiplication given by composition. The endomorphism ring
1783:
974:
567:-bimodules arise this way: other compatible right multiplications may exist.)
408:
28:
657:
1693:
1689:
1088:
1080:
404:
252:
36:
1549:
is a motivating example of a symmetric monoidal category, in which case
1718:
1699:
1092:
1744:
1706:
954:-bimodule is actually the same thing as a left module over the ring
795:
211:
16:
Abelian group equipped with compatible ring action on both sides
1084:
375:
matrix is not defined. The crucial bimodule property, that
1702:
can be seen as a categorical generalization of bimodules.
1079:
in a natural fashion. This tensor product of bimodules is
559:
has the natural structure of a bimodule. For example, if
426:-bimodule, with left and right multiplication defined by
1490:
the tensor product of the category. In particular, if
1685:
in a natural fashion. These statements extend to the
1341:
1236:
1160:
1122:
127:
1513:, which gives a monoidal embedding of the category
1113:are exactly bimodule homomorphisms, i.e. functions
1433:
1290:
1221:
1140:
178:
1781:
1765:. W. H. Freeman and Company. pp. 133–136.
1705:Note that bimodules are not at all related to
1584:giving the monoidal structure, and with unit
1757:
896:
1095:, in a canonical way – 2 morphisms between
1743:
616:. Any abelian group may be treated as a
1782:
1737:
935:if it is both a homomorphism of left
1731:
13:
1222:{\displaystyle f(m+m')=f(m)+f(m')}
678:by left multiplication defined by
500:itself can be considered to be an
14:
1801:
1291:{\displaystyle f(r.m.s)=r.f(m).s}
1575:, with the usual tensor product
1141:{\displaystyle f:M\rightarrow N}
422:has the natural structure of an
179:{\displaystyle (r.m).s=r.(m.s).}
31:that is both a left and a right
737:to itself. Therefore any right
605:-modules may be interpreted as
345:) itself is not a ring (unless
1428:
1411:
1405:
1388:
1382:
1370:
1364:
1342:
1279:
1273:
1258:
1240:
1216:
1205:
1196:
1190:
1181:
1164:
1132:
762:-bimodule. Similarly any left
697:. The bimodule property, that
481:is the canonical embedding of
197:-bimodule is also known as an
170:
158:
140:
128:
1:
1724:
1056:, then the tensor product of
42:
7:
1712:
1666:-module homomorphisms from
204:
10:
1806:
1611:-algebra. Furthermore, if
1502:-module is canonically an
733:-module homomorphism from
355:), because multiplying an
1569:category of vector spaces
1019:are valid for bimodules.
897:Further notions and facts
917:-bimodules, then a map
1498:, every left or right
1435:
1292:
1223:
1142:
939:-modules and of right
638:-module, then the set
180:
1588:. We also see that a
1436:
1293:
1224:
1143:
1064:(taken over the ring
933:bimodule homomorphism
326:matrix multiplication
181:
1339:
1234:
1158:
1120:
1017:isomorphism theorems
467:respectively, where
125:
90:-module and a right
71:is an abelian group
1087:a unique canonical
1015:, and the standard
601:. Similarly, right
1431:
1288:
1219:
1138:
552:Any module over a
365:matrix by another
176:
1469:monoidal category
407:, corresponds to
1797:
1776:
1763:Basic Algebra II
1750:
1749:
1747:
1735:
1687:derived functors
1684:
1661:
1640:
1625:
1606:
1583:
1566:
1538:. The case that
1537:
1522:
1512:
1496:commutative ring
1481:
1466:
1440:
1438:
1437:
1432:
1421:
1398:
1363:
1352:
1331:
1321:
1311:
1297:
1295:
1294:
1289:
1228:
1226:
1225:
1220:
1215:
1180:
1147:
1145:
1144:
1139:
1105:
1078:
1055:
1040:
1010:
995:
968:
930:
883:
868:
838:
827:
817:. It is also an
816:
786:
761:
725:, restates that
724:
696:
673:
651:
626:
615:
599:ring of integers
592:
554:commutative ring
548:
510:
496:is a ring, then
480:
466:
447:
402:
374:
364:
354:
319:
292:
265:
247:
185:
183:
182:
177:
78:
21:abstract algebra
1805:
1804:
1800:
1799:
1798:
1796:
1795:
1794:
1780:
1779:
1773:
1754:
1753:
1736:
1732:
1727:
1715:
1675:
1651:
1645:
1631:
1616:
1593:
1582:
1576:
1550:
1524:
1514:
1503:
1482:with the usual
1472:
1467:is exactly the
1445:
1414:
1391:
1356:
1345:
1340:
1337:
1336:
1323:
1313:
1303:
1235:
1232:
1231:
1208:
1173:
1159:
1156:
1155:
1121:
1118:
1117:
1096:
1069:
1046:
1031:
1001:
991:
982:
964:
955:
918:
899:
879:
870:
859:
829:
818:
807:
780:
771:
752:
746:
698:
679:
667:
661:
645:
639:
617:
606:
583:
539:
501:
468:
449:
427:
376:
366:
356:
346:
340:
322:matrix addition
311:
305:
284:
278:
256:
239:
233:
207:
126:
123:
122:
72:
45:
17:
12:
11:
5:
1803:
1793:
1792:
1778:
1777:
1771:
1752:
1751:
1729:
1728:
1726:
1723:
1722:
1721:
1714:
1711:
1647:
1607:is exactly an
1578:
1484:tensor product
1442:
1441:
1430:
1427:
1424:
1420:
1417:
1413:
1410:
1407:
1404:
1401:
1397:
1394:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1362:
1359:
1355:
1351:
1348:
1344:
1300:
1299:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1229:
1218:
1214:
1211:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1179:
1176:
1172:
1169:
1166:
1163:
1149:
1148:
1137:
1134:
1131:
1128:
1125:
1024:tensor product
987:
960:
898:
895:
894:
893:
875:
854:-bimodule and
840:
788:
776:
748:
663:
641:
628:
578:-module, then
568:
550:
530:Any two-sided
528:
521:direct product
490:
412:
332:
301:
293:matrices, and
274:
225:
206:
203:
187:
186:
175:
172:
169:
166:
163:
160:
157:
154:
151:
148:
145:
142:
139:
136:
133:
130:
95:
44:
41:
15:
9:
6:
4:
3:
2:
1802:
1791:
1790:Module theory
1788:
1787:
1785:
1774:
1772:0-7167-1933-9
1768:
1764:
1760:
1756:
1755:
1746:
1741:
1734:
1730:
1720:
1717:
1716:
1710:
1708:
1703:
1701:
1697:
1695:
1691:
1688:
1682:
1678:
1673:
1669:
1665:
1659:
1655:
1650:
1644:
1638:
1634:
1629:
1623:
1619:
1614:
1610:
1604:
1600:
1596:
1591:
1587:
1581:
1574:
1570:
1565:
1561:
1557:
1553:
1548:
1545:
1541:
1535:
1531:
1527:
1521:
1517:
1510:
1506:
1501:
1497:
1493:
1489:
1485:
1479:
1475:
1470:
1464:
1460:
1456:
1452:
1448:
1425:
1422:
1418:
1415:
1408:
1402:
1399:
1395:
1392:
1385:
1379:
1376:
1373:
1367:
1360:
1357:
1353:
1349:
1346:
1335:
1334:
1333:
1330:
1326:
1320:
1316:
1310:
1306:
1285:
1282:
1276:
1270:
1267:
1264:
1261:
1255:
1252:
1249:
1246:
1243:
1237:
1230:
1212:
1209:
1202:
1199:
1193:
1187:
1184:
1177:
1174:
1170:
1167:
1161:
1154:
1153:
1152:
1151:that satisfy
1135:
1129:
1126:
1123:
1116:
1115:
1114:
1112:
1108:
1103:
1099:
1094:
1090:
1086:
1082:
1076:
1072:
1067:
1063:
1059:
1053:
1049:
1044:
1038:
1034:
1029:
1025:
1020:
1018:
1014:
1008:
1004:
999:
994:
990:
985:
980:
976:
975:opposite ring
972:
967:
963:
958:
953:
949:
944:
942:
938:
934:
929:
925:
921:
916:
912:
908:
904:
891:
887:
882:
878:
873:
866:
862:
857:
853:
849:
845:
841:
836:
832:
825:
821:
814:
810:
805:
801:
797:
793:
789:
784:
779:
774:
769:
765:
760:
756:
751:
744:
740:
736:
732:
728:
722:
718:
714:
710:
706:
702:
694:
690:
686:
682:
677:
671:
666:
659:
658:endomorphisms
655:
649:
644:
637:
633:
629:
624:
620:
613:
609:
604:
600:
596:
590:
586:
581:
577:
573:
569:
566:
562:
558:
555:
551:
546:
542:
537:
533:
529:
526:
522:
518:
514:
508:
504:
499:
495:
491:
488:
484:
479:
475:
471:
464:
460:
456:
452:
446:
442:
438:
434:
430:
425:
421:
417:
413:
410:
409:associativity
406:
400:
396:
392:
388:
384:
380:
373:
369:
363:
359:
353:
349:
344:
339:
335:
331:
327:
323:
318:
314:
309:
304:
300:
296:
291:
287:
282:
277:
273:
269:
263:
259:
254:
250:
246:
242:
237:
232:
228:
224:
220:
216:
213:
210:For positive
209:
208:
202:
200:
196:
192:
173:
167:
164:
161:
155:
152:
149:
146:
143:
137:
134:
131:
120:
116:
112:
108:
104:
100:
96:
93:
89:
85:
82:
81:
80:
76:
70:
66:
62:
58:
54:
50:
40:
38:
34:
30:
29:abelian group
26:
22:
1762:
1759:Jacobson, N.
1745:math/0303175
1733:
1704:
1698:
1680:
1676:
1671:
1667:
1663:
1657:
1653:
1648:
1636:
1632:
1627:
1621:
1617:
1612:
1608:
1602:
1598:
1594:
1585:
1579:
1572:
1563:
1559:
1555:
1551:
1546:
1539:
1533:
1529:
1525:
1519:
1515:
1508:
1504:
1499:
1491:
1487:
1477:
1473:
1462:
1458:
1454:
1450:
1446:
1443:
1328:
1324:
1318:
1314:
1308:
1304:
1301:
1150:
1110:
1106:
1101:
1097:
1074:
1070:
1065:
1061:
1057:
1051:
1047:
1042:
1036:
1032:
1027:
1021:
1006:
1002:
992:
988:
983:
978:
970:
965:
961:
956:
951:
947:
945:
940:
936:
932:
927:
923:
919:
914:
910:
906:
902:
900:
889:
885:
880:
876:
871:
864:
860:
855:
851:
847:
843:
834:
830:
823:
819:
812:
808:
803:
799:
791:
782:
777:
772:
767:
763:
758:
754:
749:
742:
738:
734:
730:
726:
720:
716:
712:
708:
704:
700:
692:
688:
684:
680:
675:
669:
664:
653:
647:
642:
635:
631:
622:
618:
611:
607:
602:
594:
588:
584:
579:
575:
571:
564:
560:
556:
544:
540:
535:
524:
516:
512:
506:
502:
497:
493:
486:
482:
477:
473:
469:
462:
458:
454:
450:
444:
440:
436:
432:
428:
423:
419:
418:over a ring
415:
414:Any algebra
398:
394:
390:
386:
382:
378:
371:
367:
361:
357:
351:
347:
342:
337:
333:
329:
316:
312:
307:
302:
298:
297:is the ring
294:
289:
285:
280:
275:
271:
270:is the ring
267:
261:
257:
253:real numbers
244:
240:
235:
230:
226:
222:
218:
214:
198:
194:
190:
188:
118:
114:
110:
106:
102:
98:
91:
87:
83:
74:
68:
64:
60:
52:
48:
46:
24:
18:
1700:Profunctors
1641:, then the
1089:isomorphism
1081:associative
634:is a right
405:matrix ring
201:-bimodule.
79:such that:
37:mathematics
1725:References
1719:Profunctor
1707:bialgebras
1674:becomes a
1480:-bimodules
1104:-bimodules
1093:2-category
1009:-bimodules
943:-modules.
892:-bimodule.
787:-bimodule.
614:-bimodules
574:is a left
534:of a ring
221:, the set
86:is a left
59:, then an
43:Definition
1683:-bimodule
1639:-bimodule
1624:-bimodule
1511:-bimodule
1423:∘
1409:⊗
1400:∘
1377:⊗
1368:∘
1354:⊗
1133:→
1077:-bimodule
1054:-bimodule
1039:-bimodule
867:-bimodule
837:-bimodule
815:-bimodule
625:-bimodule
591:-bimodule
547:-bimodule
509:-bimodule
264:-bimodule
1784:Category
1761:(1989).
1713:See also
1419:′
1396:′
1361:′
1350:′
1213:′
1178:′
1068:) is an
998:category
969:, where
922: :
766:-module
741:-module
674:acts on
656:-module
593:, where
472: :
266:, where
249:matrices
212:integers
205:Examples
97:For all
94:-module.
69:bimodule
55:are two
25:bimodule
1662:of all
1013:abelian
1000:of all
973:is the
869:, then
828:and an
802:, then
796:subring
597:is the
1769:
1630:is an
1615:is an
1590:monoid
1567:, the
1322:, and
1045:is an
1030:is an
884:is an
858:is an
846:is an
806:is an
770:is an
745:is an
729:is a
582:is an
538:is an
519:-fold
255:is an
33:module
27:is an
1740:arXiv
1595:Bimod
1577:⊗ = ⊗
1571:over
1544:field
1542:is a
1526:Bimod
1523:into
1494:is a
1486:over
1455:Bimod
1085:up to
1026:: if
931:is a
794:is a
532:ideal
515:(the
485:into
310:) of
283:) of
238:) of
57:rings
1767:ISBN
1692:and
1626:and
1564:Vect
1453:) =
1302:for
1109:and
1060:and
1041:and
909:are
905:and
775:-End
448:and
324:and
217:and
113:and
77:, +)
51:and
23:, a
1694:Tor
1690:Ext
1670:to
1646:Hom
1643:set
1592:in
1556:Mod
1520:Mod
1471:of
1447:End
1011:is
977:of
946:An
901:If
842:If
798:of
790:If
747:End
662:End
652:of
640:End
630:If
570:If
523:of
492:If
251:of
189:An
117:in
109:in
101:in
47:If
19:In
1786::
1709:.
1696:.
1656:,
1601:,
1558:=
1532:,
1461:,
1327:∈
1317:∈
1312:,
1307:∈
926:→
757:)-
715:.(
711:=
707:).
687:=
527:).
476:→
459:aφ
457:=
435:=
411:).
393:.(
389:=
385:).
370:×
360:×
350:=
315:×
288:×
243:×
121::
105:,
1775:.
1748:.
1742::
1681:R
1679:-
1677:T
1672:L
1668:M
1664:S
1660:)
1658:L
1654:M
1652:(
1649:S
1637:S
1635:-
1633:T
1628:L
1622:S
1620:-
1618:R
1613:M
1609:R
1605:)
1603:R
1599:R
1597:(
1586:K
1580:K
1573:K
1562:-
1560:K
1554:-
1552:R
1547:K
1540:R
1536:)
1534:R
1530:R
1528:(
1518:-
1516:R
1509:R
1507:-
1505:R
1500:R
1492:R
1488:R
1478:R
1476:-
1474:R
1465:)
1463:R
1459:R
1457:(
1451:R
1449:(
1429:)
1426:g
1416:g
1412:(
1406:)
1403:f
1393:f
1389:(
1386:=
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