729:
3125:
2625:
434:
355:
2945:
2804:
1143:
1481:
1996:
This generalization is useful even for the study of fields – notably, many algebraic objects associated to a field are not themselves fields, but are instead rings, such as algebras over a field, as in
1710:
218:
651:
2330:
1802:
Informally, extension of scalars is "the tensor product of a ring and a module"; more formally, it is a special case of a tensor product of a bimodule and a module – the tensor product of an
3002:
2475:
2667:
3272:
3221:
646:
148:
3033:
2275:
1329:
3167:
1773:
1544:
2517:
2013:
change under extension of scalars – for example, the representation of the cyclic group of order 4, given by rotation of the plane by 90°, is an irreducible 2-dimensional
271:
2528:
767:
2202:
2055:
1240:
911:
534:
76:
1736:
1507:
1023:
997:
2150:
2057:
is irreducible of degree 2 over the reals, but factors into 2 factors of degree 1 over the complex numbers – it has no real eigenvalues, but 2 complex eigenvalues.
1836:
1627:
1596:
2017:
representation, but on extension of scalars to the complex numbers, it split into 2 complex representations of dimension 1. This corresponds to the fact that the
3025:
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2851:
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1389:
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1174:
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1031:
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2010:
3435:
2018:
1283:
1189:
1185:
3384:. Algebra I, Chapter II. LINEAR ALGEBRA.§5. Extension of the ring of scalars;§7. Vector spaces. 1974 by Hermann.
2634:
724:{\displaystyle {\begin{aligned}M\times R&\longrightarrow M\\(m,r)&\longmapsto m\cdot f(r)\end{aligned}}}
3320:
3226:
3175:
1977:
3120:{\displaystyle M\to M\otimes _{R}R{\xrightarrow {{\text{id}}_{M}\otimes f}}M\otimes _{R}S{\xrightarrow {v}}N}
104:
2227:
1945:
1288:
3140:
1898:
3325:
1998:
3430:
2520:
2006:
79:
2620:{\displaystyle M^{S}=M\otimes _{R}S{\xrightarrow {u\otimes {\text{id}}_{S}}}N_{R}\otimes _{R}S\to N}
2480:
1925:
3347:
233:
1176:
is the ring of integers, then this is just the forgetful functor from modules to abelian groups.
737:
545:
440:
1741:
1512:
3361:
3351:
2175:
2024:
1213:
884:
507:
49:
44:
3381:
3339:
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1851:
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1486:
1002:
976:
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501:
497:
28:
2128:
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1569:
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956:
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792:
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603:
583:
563:
479:
459:
17:
3419:
3132:
2807:
1150:
3376:
3372:
3303:
1882:
2001:. Just as one can extend scalars on vector spaces, one can also extend scalars on
3135:
1855:
3295:
2002:
1902:
429:{\displaystyle f^{*}:{\text{Mod}}_{S}\leftrightarrows {\text{Mod}}_{R}:f_{*}.}
3365:
350:{\displaystyle f_{!}:{\text{Mod}}_{R}\leftrightarrows {\text{Mod}}_{S}:f^{*}}
3172:
This construction establishes a one to one correspondence between the sets
1956:-algebra). For example, the result of complexifying a real vector space (
1599:
1411:
is also a right module over itself, and the two actions commute, that is
2940:{\displaystyle G:{\text{Hom}}_{S}(M^{S},N)\to {\text{Hom}}_{R}(M,N_{R})}
2799:{\displaystyle F:{\text{Hom}}_{R}(M,N_{R})\to {\text{Hom}}_{S}(M^{S},N)}
2336:
Relation between the extension of scalars and the restriction of scalars
1138:{\displaystyle u(m\cdot r)=u(m\cdot f(r))=u(m)\cdot f(r)=u(m)\cdot r\,}
818:
3107:
3064:
2566:
1936:-module, the resulting module can be thought of alternatively as an
548:, the term "restriction of scalars" is often used as a synonym for
3394:
3274:. Actually, this correspondence depends only on the homomorphism
3346:. Foote, Richard M. (3 ed.). Hoboken, NJ: Wiley. pp.
1885:, and thus extension of scalars converts a vector space over
1881:
In the language of fields, a module over a field is called a
78:, there are three ways to change the coefficient ring of a
35:
is an operation of changing a coefficient ring to another.
2065:
Extension of scalars can be interpreted as a functor from
1972:) can be interpreted either as a complex vector space (
3280:
3229:
3178:
3143:
3036:
3010:
2953:
2859:
2839:
2819:
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2698:
2675:
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2483:
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2230:
2210:
2178:
2158:
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2111:
2091:
2071:
2027:
1908:
More generally, given a homomorphism from a field or
1812:
1781:
1744:
1718:
1655:
1635:
1608:
1572:
1552:
1515:
1489:
1476:{\displaystyle r\cdot (s\cdot s')=(r\cdot s)\cdot s'}
1417:
1397:
1377:
1357:
1337:
1291:
1268:
1248:
1216:
1190:
Tensor product of modules § Extension of scalars
1162:
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1005:
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462:
369:
293:
236:
165:
107:
52:
213:{\displaystyle f_{*}M=\operatorname {Hom} _{R}(S,M)}
2853:have an identity, there is an inverse homomorphism
3286:
3266:
3215:
3161:
3119:
3019:
2996:
2939:
2845:
2825:
2798:
2704:
2684:
2661:
2619:
2511:
2469:
2412:
2392:
2372:
2352:
2325:{\displaystyle u^{S}=u\otimes _{R}{\text{id}}_{S}}
2324:
2269:
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2164:
2144:
2117:
2097:
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2049:
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1234:
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925:
905:
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853:
833:
801:
781:
761:
723:
632:
612:
592:
572:
536:be a homomorphism. Restriction of scalars changes
528:
488:
468:
428:
349:
265:
212:
142:
70:
1705:{\displaystyle (m\otimes s)\cdot s'=m\otimes ss'}
2005:and also on modules over group algebras, i.e.,
2997:{\displaystyle v\in {\text{Hom}}_{S}(M^{S},N)}
2470:{\displaystyle u\in {\text{Hom}}_{R}(M,N_{R})}
1242:be a homomorphism between two rings, and let
8:
3395:Induction and Coinduction of Representations
1149:As a functor, restriction of scalars is the
2806:is well-defined, and is a homomorphism (of
600:. Then it can be regarded as a module over
2662:{\displaystyle n\otimes s\mapsto n\cdot s}
1976:-module) or as a real vector space with a
1775:. This module is said to be obtained from
933:-homomorphism between the restrictions of
817:Restriction of scalars can be viewed as a
3279:
3267:{\displaystyle {\text{Hom}}_{R}(M,N_{R})}
3255:
3236:
3231:
3228:
3216:{\displaystyle {\text{Hom}}_{S}(M^{S},N)}
3198:
3185:
3180:
3177:
3142:
3102:
3093:
3071:
3066:
3059:
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2979:
2966:
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2885:
2872:
2867:
2858:
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2818:
2781:
2768:
2763:
2750:
2731:
2726:
2717:
2697:
2674:
2636:
2602:
2592:
2579:
2574:
2561:
2552:
2536:
2530:
2497:
2482:
2458:
2439:
2434:
2425:
2405:
2385:
2365:
2345:
2316:
2311:
2304:
2288:
2282:
2261:
2248:
2235:
2229:
2209:
2177:
2157:
2136:
2130:
2110:
2090:
2070:
2032:
2026:
1854:, which is extension of scalars from the
1811:
1780:
1743:
1717:
1654:
1634:
1613:
1607:
1571:
1551:
1514:
1488:
1416:
1396:
1376:
1356:
1336:
1312:
1296:
1290:
1267:
1247:
1215:
1161:
1134:
1033:
1004:
978:
958:
938:
918:
886:
866:
846:
826:
794:
774:
739:
650:
648:
625:
605:
585:
565:
509:
481:
461:
417:
404:
399:
389:
384:
374:
368:
341:
328:
323:
313:
308:
298:
292:
257:
241:
235:
186:
170:
164:
131:
112:
106:
51:
1932:and thus when one extends scalars on an
3400:
3306:to the restriction of scalars functor.
3407:
3302:, the extension of scalars functor is
2009:. Particularly useful is relating how
1153:of the extension of scalars functor.
496:be two rings (they may or may not be
143:{\displaystyle f_{!}M=M\otimes _{R}S}
7:
2270:{\displaystyle u^{S}:M^{S}\to N^{S}}
1324:{\displaystyle M^{S}=M\otimes _{R}S}
3162:{\displaystyle m\mapsto m\otimes 1}
2947:, which is defined as follows. Let
769:denotes the action defined by the
25:
1901:(extension from the reals to the
1850:One of the simplest examples is
3261:
3242:
3210:
3191:
3147:
3040:
2991:
2972:
2934:
2915:
2900:
2897:
2878:
2793:
2774:
2759:
2756:
2737:
2647:
2611:
2503:
2464:
2445:
2254:
2188:
1825:
1813:
1668:
1656:
1585:
1573:
1459:
1447:
1441:
1424:
1226:
1125:
1119:
1110:
1104:
1095:
1089:
1080:
1077:
1071:
1059:
1050:
1038:
897:
756:
750:
714:
708:
696:
689:
677:
667:
520:
395:
319:
207:
195:
62:
1:
2512:{\displaystyle Fu:M^{S}\to N}
1194:Extension of scalars changes
456:Throughout this section, let
1873:one can extend scalars from
1862:. More generally, given any
1546:(in a more formal language,
266:{\displaystyle f^{*}N=N_{R}}
3131:where the first map is the
2061:Interpretation as a functor
2011:irreducible representations
1980:(algebra representation of
1629:inherits a right action of
813:Interpretation as a functor
762:{\displaystyle m\cdot f(r)}
3452:
1893:This can also be done for
1183:
2712:-homomorphism, and hence
2019:characteristic polynomial
1768:{\displaystyle s,s'\in S}
1539:{\displaystyle s,s'\in S}
1186:Tensor product of modules
913:automatically becomes an
3321:Tensor product of fields
2197:{\displaystyle u:M\to N}
2050:{\displaystyle x^{2}+1,}
1978:linear complex structure
1924:can be thought of as an
1235:{\displaystyle f:R\to S}
906:{\displaystyle u:M\to N}
529:{\displaystyle f:R\to S}
226:co-extension of scalars,
71:{\displaystyle f:R\to S}
2420:. Given a homomorphism
1889:to a vector space over
275:restriction of scalars.
3338:Dummit, David (2004).
3288:
3268:
3217:
3163:
3121:
3021:
2998:
2941:
2847:
2827:
2800:
2706:
2686:
2663:
2631:where the last map is
2621:
2513:
2471:
2414:
2394:
2374:
2354:
2326:
2271:
2218:
2198:
2166:
2146:
2119:
2099:
2079:
2051:
1946:algebra representation
1832:
1789:
1769:
1732:
1731:{\displaystyle m\in M}
1706:
1643:
1623:
1592:
1560:
1540:
1503:
1502:{\displaystyle r\in R}
1477:
1405:
1385:
1365:
1351:is regarded as a left
1345:
1325:
1276:
1256:
1236:
1170:
1139:
1019:
1018:{\displaystyle r\in R}
993:
992:{\displaystyle m\in M}
967:
947:
927:
907:
875:
855:
835:
803:
783:
763:
725:
634:
614:
594:
574:
530:
490:
470:
452:Restriction of scalars
430:
351:
267:
214:
144:
82:; namely, for a right
72:
18:Restriction of scalars
3326:Tensor-hom adjunction
3298:. In the language of
3289:
3269:
3218:
3164:
3122:
3022:
2999:
2942:
2848:
2828:
2801:
2707:
2687:
2664:
2622:
2514:
2472:
2415:
2395:
2375:
2355:
2327:
2272:
2219:
2199:
2167:
2147:
2145:{\displaystyle M^{S}}
2120:
2100:
2080:
2052:
2007:group representations
1999:representation theory
1833:
1831:{\displaystyle (R,S)}
1790:
1770:
1733:
1707:
1644:
1624:
1622:{\displaystyle M^{S}}
1593:
1591:{\displaystyle (R,S)}
1561:
1541:
1504:
1478:
1406:
1386:
1366:
1346:
1326:
1277:
1257:
1237:
1171:
1140:
1020:
994:
968:
948:
928:
908:
876:
856:
836:
804:
789:-module structure on
784:
764:
726:
635:
615:
595:
575:
531:
491:
471:
431:
352:
268:
215:
156:extension of scalars,
145:
73:
3377:Notes on Tor and Ext
3278:
3227:
3176:
3141:
3034:
3008:
2951:
2857:
2837:
2817:
2716:
2696:
2673:
2635:
2529:
2481:
2424:
2404:
2384:
2364:
2344:
2281:
2228:
2208:
2176:
2156:
2129:
2109:
2089:
2069:
2025:
1810:
1797:extension of scalars
1779:
1742:
1716:
1653:
1633:
1606:
1570:
1550:
1513:
1487:
1415:
1395:
1375:
1355:
1335:
1289:
1266:
1246:
1214:
1180:Extension of scalars
1160:
1032:
1003:
977:
957:
937:
917:
885:
865:
845:
825:
793:
773:
738:
647:
624:
620:where the action of
604:
584:
564:
508:
480:
460:
367:
291:
280:They are related as
234:
163:
105:
50:
3426:Commutative algebra
3111:
3083:
3027:is the composition
2585:
2152:, as above, and an
2105:-modules. It sends
1926:associative algebra
1899:quaternionification
439:This is related to
3284:
3264:
3213:
3159:
3117:
3020:{\displaystyle Gv}
3017:
2994:
2937:
2843:
2823:
2796:
2702:
2685:{\displaystyle Fu}
2682:
2659:
2617:
2509:
2467:
2410:
2390:
2370:
2350:
2322:
2267:
2214:
2194:
2162:
2142:
2115:
2095:
2075:
2047:
2021:of this operator,
1940:-module, or as an
1828:
1785:
1765:
1728:
1702:
1639:
1619:
1588:
1556:
1536:
1499:
1473:
1401:
1381:
1361:
1341:
1321:
1272:
1252:
1232:
1166:
1135:
1015:
989:
963:
943:
923:
903:
871:
851:
831:
799:
779:
759:
721:
719:
630:
610:
590:
570:
546:algebraic geometry
526:
486:
466:
426:
347:
263:
210:
140:
68:
3287:{\displaystyle f}
3234:
3183:
3112:
3084:
3069:
2964:
2907:
2870:
2846:{\displaystyle S}
2826:{\displaystyle R}
2766:
2729:
2705:{\displaystyle S}
2586:
2577:
2437:
2413:{\displaystyle N}
2393:{\displaystyle S}
2373:{\displaystyle M}
2353:{\displaystyle R}
2314:
2217:{\displaystyle S}
2165:{\displaystyle R}
2118:{\displaystyle M}
2098:{\displaystyle S}
2078:{\displaystyle R}
1895:division algebras
1788:{\displaystyle M}
1649:. It is given by
1642:{\displaystyle S}
1559:{\displaystyle S}
1404:{\displaystyle S}
1384:{\displaystyle f}
1364:{\displaystyle R}
1344:{\displaystyle S}
1275:{\displaystyle R}
1262:be a module over
1255:{\displaystyle M}
1169:{\displaystyle R}
966:{\displaystyle N}
946:{\displaystyle M}
926:{\displaystyle R}
874:{\displaystyle S}
854:{\displaystyle R}
834:{\displaystyle S}
802:{\displaystyle M}
782:{\displaystyle S}
633:{\displaystyle R}
613:{\displaystyle R}
593:{\displaystyle S}
580:is a module over
573:{\displaystyle M}
489:{\displaystyle S}
469:{\displaystyle R}
402:
387:
326:
311:
222:coinduced module,
45:ring homomorphism
16:(Redirected from
3443:
3436:Adjoint functors
3411:
3405:
3382:Nicolas Bourbaki
3369:
3345:
3342:Abstract algebra
3293:
3291:
3290:
3285:
3273:
3271:
3270:
3265:
3260:
3259:
3241:
3240:
3235:
3232:
3222:
3220:
3219:
3214:
3203:
3202:
3190:
3189:
3184:
3181:
3168:
3166:
3165:
3160:
3126:
3124:
3123:
3118:
3113:
3103:
3098:
3097:
3085:
3076:
3075:
3070:
3067:
3060:
3055:
3054:
3026:
3024:
3023:
3018:
3003:
3001:
3000:
2995:
2984:
2983:
2971:
2970:
2965:
2962:
2946:
2944:
2943:
2938:
2933:
2932:
2914:
2913:
2908:
2905:
2890:
2889:
2877:
2876:
2871:
2868:
2852:
2850:
2849:
2844:
2832:
2830:
2829:
2824:
2805:
2803:
2802:
2797:
2786:
2785:
2773:
2772:
2767:
2764:
2755:
2754:
2736:
2735:
2730:
2727:
2711:
2709:
2708:
2703:
2691:
2689:
2688:
2683:
2668:
2666:
2665:
2660:
2626:
2624:
2623:
2618:
2607:
2606:
2597:
2596:
2587:
2584:
2583:
2578:
2575:
2562:
2557:
2556:
2541:
2540:
2518:
2516:
2515:
2510:
2502:
2501:
2476:
2474:
2473:
2468:
2463:
2462:
2444:
2443:
2438:
2435:
2419:
2417:
2416:
2411:
2399:
2397:
2396:
2391:
2379:
2377:
2376:
2371:
2359:
2357:
2356:
2351:
2331:
2329:
2328:
2323:
2321:
2320:
2315:
2312:
2309:
2308:
2293:
2292:
2276:
2274:
2273:
2268:
2266:
2265:
2253:
2252:
2240:
2239:
2223:
2221:
2220:
2215:
2203:
2201:
2200:
2195:
2171:
2169:
2168:
2163:
2151:
2149:
2148:
2143:
2141:
2140:
2124:
2122:
2121:
2116:
2104:
2102:
2101:
2096:
2084:
2082:
2081:
2076:
2056:
2054:
2053:
2048:
2037:
2036:
1944:-module with an
1897:, as is done in
1869: <
1852:complexification
1838:-bimodule is an
1837:
1835:
1834:
1829:
1806:-module with an
1794:
1792:
1791:
1786:
1774:
1772:
1771:
1766:
1758:
1737:
1735:
1734:
1729:
1711:
1709:
1708:
1703:
1701:
1681:
1648:
1646:
1645:
1640:
1628:
1626:
1625:
1620:
1618:
1617:
1597:
1595:
1594:
1589:
1565:
1563:
1562:
1557:
1545:
1543:
1542:
1537:
1529:
1508:
1506:
1505:
1500:
1482:
1480:
1479:
1474:
1472:
1440:
1410:
1408:
1407:
1402:
1390:
1388:
1387:
1382:
1370:
1368:
1367:
1362:
1350:
1348:
1347:
1342:
1330:
1328:
1327:
1322:
1317:
1316:
1301:
1300:
1281:
1279:
1278:
1273:
1261:
1259:
1258:
1253:
1241:
1239:
1238:
1233:
1175:
1173:
1172:
1167:
1144:
1142:
1141:
1136:
1024:
1022:
1021:
1016:
998:
996:
995:
990:
972:
970:
969:
964:
952:
950:
949:
944:
932:
930:
929:
924:
912:
910:
909:
904:
880:
878:
877:
872:
860:
858:
857:
852:
840:
838:
837:
832:
808:
806:
805:
800:
788:
786:
785:
780:
768:
766:
765:
760:
730:
728:
727:
722:
720:
639:
637:
636:
631:
619:
617:
616:
611:
599:
597:
596:
591:
579:
577:
576:
571:
550:Weil restriction
535:
533:
532:
527:
500:, or contain an
495:
493:
492:
487:
475:
473:
472:
467:
435:
433:
432:
427:
422:
421:
409:
408:
403:
400:
394:
393:
388:
385:
379:
378:
356:
354:
353:
348:
346:
345:
333:
332:
327:
324:
318:
317:
312:
309:
303:
302:
282:adjoint functors
272:
270:
269:
264:
262:
261:
246:
245:
219:
217:
216:
211:
191:
190:
175:
174:
149:
147:
146:
141:
136:
135:
117:
116:
77:
75:
74:
69:
21:
3451:
3450:
3446:
3445:
3444:
3442:
3441:
3440:
3416:
3415:
3414:
3406:
3402:
3391:
3389:Further reading
3358:
3337:
3334:
3312:
3300:category theory
3276:
3275:
3251:
3230:
3225:
3224:
3194:
3179:
3174:
3173:
3139:
3138:
3089:
3065:
3046:
3032:
3031:
3006:
3005:
2975:
2960:
2949:
2948:
2924:
2903:
2881:
2866:
2855:
2854:
2835:
2834:
2815:
2814:
2777:
2762:
2746:
2725:
2714:
2713:
2694:
2693:
2671:
2670:
2633:
2632:
2598:
2588:
2573:
2548:
2532:
2527:
2526:
2493:
2479:
2478:
2454:
2433:
2422:
2421:
2402:
2401:
2382:
2381:
2362:
2361:
2342:
2341:
2338:
2310:
2300:
2284:
2279:
2278:
2257:
2244:
2231:
2226:
2225:
2206:
2205:
2174:
2173:
2154:
2153:
2132:
2127:
2126:
2107:
2106:
2087:
2086:
2067:
2066:
2063:
2028:
2023:
2022:
1994:
1864:field extension
1860:complex numbers
1848:
1808:
1807:
1777:
1776:
1751:
1740:
1739:
1714:
1713:
1694:
1674:
1651:
1650:
1631:
1630:
1609:
1604:
1603:
1568:
1567:
1548:
1547:
1522:
1511:
1510:
1485:
1484:
1465:
1433:
1413:
1412:
1393:
1392:
1373:
1372:
1353:
1352:
1333:
1332:
1308:
1292:
1287:
1286:
1282:. Consider the
1264:
1263:
1244:
1243:
1212:
1211:
1208:
1192:
1182:
1158:
1157:
1030:
1029:
1001:
1000:
975:
974:
955:
954:
935:
934:
915:
914:
883:
882:
863:
862:
843:
842:
823:
822:
815:
791:
790:
771:
770:
736:
735:
718:
717:
692:
674:
673:
663:
645:
644:
622:
621:
602:
601:
582:
581:
562:
561:
558:
506:
505:
478:
477:
458:
457:
454:
449:
441:Shapiro's lemma
413:
398:
383:
370:
365:
364:
337:
322:
307:
294:
289:
288:
253:
237:
232:
231:
182:
166:
161:
160:
152:induced module,
127:
108:
103:
102:
98:, one can form
48:
47:
41:
33:change of rings
23:
22:
15:
12:
11:
5:
3449:
3447:
3439:
3438:
3433:
3428:
3418:
3417:
3413:
3412:
3410:, p. 359.
3399:
3398:
3397:
3390:
3387:
3386:
3385:
3379:
3370:
3356:
3333:
3330:
3329:
3328:
3323:
3318:
3316:Six operations
3311:
3308:
3283:
3263:
3258:
3254:
3250:
3247:
3244:
3239:
3212:
3209:
3206:
3201:
3197:
3193:
3188:
3158:
3155:
3152:
3149:
3146:
3129:
3128:
3116:
3110:
3106:
3101:
3096:
3092:
3088:
3082:
3079:
3074:
3063:
3058:
3053:
3049:
3045:
3042:
3039:
3016:
3013:
2993:
2990:
2987:
2982:
2978:
2974:
2969:
2959:
2956:
2936:
2931:
2927:
2923:
2920:
2917:
2912:
2902:
2899:
2896:
2893:
2888:
2884:
2880:
2875:
2865:
2862:
2842:
2822:
2808:abelian groups
2795:
2792:
2789:
2784:
2780:
2776:
2771:
2761:
2758:
2753:
2749:
2745:
2742:
2739:
2734:
2724:
2721:
2701:
2681:
2678:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2629:
2628:
2616:
2613:
2610:
2605:
2601:
2595:
2591:
2582:
2572:
2569:
2565:
2560:
2555:
2551:
2547:
2544:
2539:
2535:
2508:
2505:
2500:
2496:
2492:
2489:
2486:
2466:
2461:
2457:
2453:
2450:
2447:
2442:
2432:
2429:
2409:
2389:
2369:
2349:
2337:
2334:
2319:
2307:
2303:
2299:
2296:
2291:
2287:
2264:
2260:
2256:
2251:
2247:
2243:
2238:
2234:
2224:-homomorphism
2213:
2193:
2190:
2187:
2184:
2181:
2172:-homomorphism
2161:
2139:
2135:
2114:
2094:
2074:
2062:
2059:
2046:
2043:
2040:
2035:
2031:
2003:group algebras
1993:
1990:
1847:
1844:
1827:
1824:
1821:
1818:
1815:
1784:
1764:
1761:
1757:
1754:
1750:
1747:
1727:
1724:
1721:
1700:
1697:
1693:
1690:
1687:
1684:
1680:
1677:
1673:
1670:
1667:
1664:
1661:
1658:
1638:
1616:
1612:
1587:
1584:
1581:
1578:
1575:
1555:
1535:
1532:
1528:
1525:
1521:
1518:
1498:
1495:
1492:
1471:
1468:
1464:
1461:
1458:
1455:
1452:
1449:
1446:
1443:
1439:
1436:
1432:
1429:
1426:
1423:
1420:
1400:
1380:
1360:
1340:
1320:
1315:
1311:
1307:
1304:
1299:
1295:
1284:tensor product
1271:
1251:
1231:
1228:
1225:
1222:
1219:
1207:
1204:
1198:-modules into
1181:
1178:
1165:
1147:
1146:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1014:
1011:
1008:
988:
985:
982:
962:
942:
922:
902:
899:
896:
893:
890:
881:-homomorphism
870:
850:
830:
814:
811:
798:
778:
758:
755:
752:
749:
746:
743:
732:
731:
716:
713:
710:
707:
704:
701:
698:
695:
693:
691:
688:
685:
682:
679:
676:
675:
672:
669:
666:
664:
662:
659:
656:
653:
652:
629:
609:
589:
569:
557:
554:
540:-modules into
525:
522:
519:
516:
513:
485:
465:
453:
450:
448:
445:
437:
436:
425:
420:
416:
412:
407:
397:
392:
382:
377:
373:
358:
357:
344:
340:
336:
331:
321:
316:
306:
301:
297:
278:
277:
260:
256:
252:
249:
244:
240:
229:
209:
206:
203:
200:
197:
194:
189:
185:
181:
178:
173:
169:
158:
139:
134:
130:
126:
123:
120:
115:
111:
67:
64:
61:
58:
55:
40:
37:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3448:
3437:
3434:
3432:
3429:
3427:
3424:
3423:
3421:
3409:
3404:
3401:
3396:
3393:
3392:
3388:
3383:
3380:
3378:
3374:
3371:
3367:
3363:
3359:
3353:
3349:
3344:
3343:
3336:
3335:
3331:
3327:
3324:
3322:
3319:
3317:
3314:
3313:
3309:
3307:
3305:
3301:
3297:
3281:
3256:
3252:
3248:
3245:
3237:
3207:
3204:
3199:
3195:
3186:
3170:
3156:
3153:
3150:
3144:
3137:
3134:
3114:
3108:
3104:
3099:
3094:
3090:
3086:
3080:
3077:
3072:
3061:
3056:
3051:
3047:
3043:
3037:
3030:
3029:
3028:
3014:
3011:
2988:
2985:
2980:
2976:
2967:
2957:
2954:
2929:
2925:
2921:
2918:
2910:
2894:
2891:
2886:
2882:
2873:
2863:
2860:
2840:
2820:
2813:In case both
2811:
2809:
2790:
2787:
2782:
2778:
2769:
2751:
2747:
2743:
2740:
2732:
2722:
2719:
2699:
2679:
2676:
2656:
2653:
2650:
2644:
2641:
2638:
2614:
2608:
2603:
2599:
2593:
2589:
2580:
2570:
2567:
2563:
2558:
2553:
2549:
2545:
2542:
2537:
2533:
2525:
2524:
2523:
2522:
2506:
2498:
2494:
2490:
2487:
2484:
2459:
2455:
2451:
2448:
2440:
2430:
2427:
2407:
2387:
2367:
2347:
2335:
2333:
2317:
2305:
2301:
2297:
2294:
2289:
2285:
2262:
2258:
2249:
2245:
2241:
2236:
2232:
2211:
2191:
2185:
2182:
2179:
2159:
2137:
2133:
2112:
2092:
2072:
2060:
2058:
2044:
2041:
2038:
2033:
2029:
2020:
2016:
2012:
2008:
2004:
2000:
1991:
1989:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1923:
1919:
1915:
1911:
1906:
1904:
1900:
1896:
1892:
1888:
1884:
1880:
1876:
1872:
1868:
1865:
1861:
1857:
1853:
1845:
1843:
1841:
1822:
1819:
1816:
1805:
1800:
1798:
1782:
1762:
1759:
1755:
1752:
1748:
1745:
1725:
1722:
1719:
1698:
1695:
1691:
1688:
1685:
1682:
1678:
1675:
1671:
1665:
1662:
1659:
1636:
1614:
1610:
1601:
1582:
1579:
1576:
1553:
1533:
1530:
1526:
1523:
1519:
1516:
1496:
1493:
1490:
1469:
1466:
1462:
1456:
1453:
1450:
1444:
1437:
1434:
1430:
1427:
1421:
1418:
1398:
1378:
1358:
1338:
1318:
1313:
1309:
1305:
1302:
1297:
1293:
1285:
1269:
1249:
1229:
1223:
1220:
1217:
1205:
1203:
1201:
1197:
1191:
1187:
1179:
1177:
1163:
1154:
1152:
1151:right adjoint
1131:
1128:
1122:
1116:
1113:
1107:
1101:
1098:
1092:
1086:
1083:
1074:
1068:
1065:
1062:
1056:
1053:
1047:
1044:
1041:
1035:
1028:
1027:
1026:
1012:
1009:
1006:
986:
983:
980:
973:. Indeed, if
960:
940:
920:
900:
894:
891:
888:
868:
861:-modules. An
848:
828:
820:
812:
810:
796:
776:
753:
747:
744:
741:
711:
705:
702:
699:
694:
686:
683:
680:
670:
665:
660:
657:
654:
643:
642:
641:
640:is given via
627:
607:
587:
567:
560:Suppose that
555:
553:
551:
547:
544:-modules. In
543:
539:
523:
517:
514:
511:
503:
499:
483:
463:
451:
446:
444:
442:
423:
418:
414:
410:
405:
390:
380:
375:
371:
363:
362:
361:
342:
338:
334:
329:
314:
304:
299:
295:
287:
286:
285:
283:
276:
258:
254:
250:
247:
242:
238:
230:
227:
223:
204:
201:
198:
192:
187:
183:
179:
176:
171:
167:
159:
157:
153:
137:
132:
128:
124:
121:
118:
113:
109:
101:
100:
99:
97:
93:
89:
85:
81:
65:
59:
56:
53:
46:
39:Constructions
38:
36:
34:
30:
19:
3403:
3373:J. Peter May
3341:
3304:left adjoint
3294:, and so is
3171:
3130:
2812:
2630:
2340:Consider an
2339:
2085:-modules to
2064:
2014:
1995:
1992:Applications
1985:
1981:
1973:
1969:
1965:
1961:
1957:
1953:
1949:
1941:
1937:
1933:
1929:
1921:
1917:
1913:
1909:
1907:
1890:
1886:
1883:vector space
1878:
1874:
1870:
1866:
1856:real numbers
1849:
1839:
1803:
1801:
1796:
1371:-module via
1209:
1199:
1195:
1193:
1155:
1148:
841:-modules to
816:
733:
559:
541:
537:
455:
438:
359:
279:
274:
273:, formed by
225:
221:
155:
151:
95:
91:
90:and a right
87:
83:
42:
32:
26:
3431:Ring theory
3408:Dummit 2004
3136:isomorphism
2521:composition
2277:defined by
1910:commutative
1903:quaternions
504:), and let
498:commutative
3420:Categories
3357:0471452343
3332:References
3296:functorial
2519:to be the
1988:-module).
1916:to a ring
1206:Definition
1202:-modules.
1184:See also:
556:Definition
447:Operations
224:formed by
154:formed by
3366:248917264
3154:⊗
3148:↦
3133:canonical
3091:⊗
3078:⊗
3048:⊗
3041:→
2958:∈
2901:→
2760:→
2654:⋅
2648:↦
2642:⊗
2612:→
2600:⊗
2571:⊗
2550:⊗
2504:→
2477:, define
2431:∈
2302:⊗
2255:→
2189:→
1920:the ring
1842:-module.
1760:∈
1723:∈
1689:⊗
1672:⋅
1663:⊗
1531:∈
1494:∈
1463:⋅
1454:⋅
1431:⋅
1422:⋅
1310:⊗
1227:→
1129:⋅
1099:⋅
1066:⋅
1045:⋅
1010:∈
984:∈
898:→
745:⋅
703:⋅
697:⟼
668:⟶
658:×
521:→
419:∗
396:⇆
376:∗
343:∗
320:⇆
243:∗
193:
172:∗
129:⊗
63:→
3310:See also
3105:→
3062:→
2564:→
2400:-module
2360:-module
1846:Examples
1795:through
1756:′
1699:′
1679:′
1600:bimodule
1527:′
1470:′
1438:′
1391:. Since
1331:, where
502:identity
94:-module
86:-module
43:Given a
3004:. Then
2669:. This
2380:and an
2204:to the
1952:(as an
1858:to the
1025:, then
819:functor
29:algebra
3364:
3354:
3350:–377.
2692:is an
1984:as an
734:where
220:, the
150:, the
80:module
1928:over
1912:ring
1566:is a
821:from
3362:OCLC
3352:ISBN
3223:and
2833:and
2015:real
1712:for
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