1466:
190:
without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain. Some authors use this as the definition of a torsion element, but this definition does not work well over more general
728:
1370:
1435:
786:
1276:
1052:
929:
1094:
can be interpreted as the set of the elements that "vanish in the localization". The same interpretation continues to hold in the non-commutative setting for rings satisfying the
839:
in two variables. For non-finitely generated modules, the above direct decomposition is not true. The torsion subgroup of an abelian group may not be a
1285:
1676:
1643:
492:, conversely, asks whether a finitely generated periodic group must be finite. The answer is "no" in general, even if the period is fixed.
1586:
1449:
reflects this relation with the algebraic torsion. This same result holds for non-commutative rings as well as long as the set
686:-module in the natural way, then (as a result of many things, either simply by finite-dimensionality or as a consequence of the
107:(in fact, this is the origin of the terminology, which was introduced for abelian groups before being generalized to modules).
1375:
295:
1695:
1668:
1616:
1542:
157:
1705:
687:
1611:
745:
1215:
1537:
649:
531:
1517:
720:
1527:
1522:
1001:
984:
667:
485:
31:
1532:
885:
1700:
1058:
712:
527:
1606:
489:
1490:
1210:
939:
519:
496:
276:
119:
85:
50:
1507:
1202:
1122:
836:
467:
357:
208:
111:
96:
78:
69:
formed by the torsion elements (in cases when this is indeed a submodule, such as when the ring is
1602:
951:
868:
602:
459:
146:
58:
1595:
1578:
1672:
1639:
1582:
265:
1631:
1591:
1502:
968:
621:
384:
187:
139:
70:
54:
1653:
1649:
1474:
660:
566:
273:
183:
17:
103:"). This is allowed by the fact that the abelian groups are the modules over the ring of
1486:
1465:
840:
593:
in one variable is pure torsion. Both these examples can be generalized as follows: if
1689:
1623:
1454:
1099:
1095:
653:
507:
500:
414:
401:
249:
123:
92:
569:
modulo 1, is periodic, i.e. every element has finite order. Analogously, the module
1660:
481:
463:
164:
1512:
1146:
800:
736:
523:
435:
42:
38:
88:, that is, when the regular elements of the ring are all its nonzero elements.
1635:
729:
structure theorem for finitely generated modules over a principal ideal domain
590:
816:
66:
1365:{\displaystyle 0\to \operatorname {Tor} _{1}^{R}(M,R_{S}/R)\to M\to M_{S}}
367:
of the group if it has finite order, i.e., if there is a positive integer
216:
is commutative then the set of all torsion elements forms a submodule of
100:
1630:, Problem Books in Mathematics, New York: Springer, pp. xviii+412,
1446:
950:
is a vector space, possibly infinite-dimensional. There is a canonical
104:
542:. In this case, torsion elements do not form a subgroup, for example,
1469:
The 4-torsion subgroup of an elliptic curve over the complex numbers.
450:
is not a domain then torsion is considered with respect to the set
421:
of integers, and in this case the two notions of torsion coincide.
1464:
77:
is a module consisting entirely of torsion elements. A module is
126:
case, the torsion elements do not form a subgroup, in general.
652:
by a subgroup is torsion-free exactly when the subgroup is a
530:, any nontrivial torsion element either has order two and is
163:
of the ring (an element that is neither a left nor a right
1579:
Introduction to
Commutative algebra and algebraic geometry
971:
of this homomorphism is precisely the torsion submodule T(
84:
This terminology is more commonly used for modules over a
413:
if its only torsion element is the identity element. Any
442:. Then it follows immediately from the definitions that
1430:{\displaystyle \operatorname {Tor} _{1}^{R}(M,R_{S}/R)}
1378:
1288:
1218:
1004:
888:
748:
827:
hold for more general commutative domains, even for
538:or has order three and is conjugate to the element
1429:
1364:
1270:
1121:The concept of torsion plays an important role in
1046:
923:
819:, any finitely generated torsion-free module over
780:
240:) may or may not be a submodule. It is shown in (
212:if zero is the only torsion element. If the ring
81:if its only torsion element is the zero element.
979:is a multiplicatively closed subset of the ring
781:{\displaystyle M\simeq F\oplus \mathrm {T} (M),}
405:if all its elements are torsion elements, and a
1271:{\displaystyle 0\to R\to R_{S}\to R_{S}/R\to 0}
206:if all its elements are torsion elements, and
473:is torsion-free when viewed as a module over
8:
314:-torsion element if there exists an element
95:(with "module" and "submodule" replaced by "
731:gives a detailed description of the module
1133:are two modules over a commutative domain
648:, +) are torsion-free. The quotient of a
1671:(Third ed.), Springer, p. 446,
1437:is the kernel of the localisation map of
1416:
1410:
1388:
1383:
1377:
1356:
1332:
1326:
1304:
1299:
1287:
1254:
1248:
1235:
1217:
1047:{\displaystyle M_{S}=M\otimes _{R}R_{S},}
1035:
1025:
1009:
1003:
909:
893:
887:
761:
747:
53:that yields zero when multiplied by some
417:may be viewed as a module over the ring
345:the set of regular elements of the ring
1554:
1137:(for example, two abelian groups, when
1561:
924:{\displaystyle M_{Q}=M\otimes _{R}Q,}
7:
268:are Ore, this covers the case when
241:
1282:-modules yields an exact sequence
762:
349:and recover the definition above.
279:(which might not be commutative).
25:
1489:they may be computed in terms of
484:(abelian or not) is periodic and
1598:", University of Michigan, 1954.
1180:is canonically isomorphic to Tor
1090:. Thus the torsion submodule of
1082:, whose kernel is precisely the
1069:. There is a canonical map from
739:. In particular, it claims that
703:Case of a principal ideal domain
480:By contrast with example 1, any
341:In particular, one can take for
156:of the module if there exists a
1628:Exercises in modules and rings
1424:
1397:
1349:
1343:
1340:
1313:
1292:
1262:
1241:
1228:
1222:
1117:Torsion in homological algebra
811:) is the torsion submodule of
772:
766:
514:obtained from the group SL(2,
296:multiplicatively closed subset
1:
1669:Graduate Texts in Mathematics
1543:Universal coefficient theorem
1481:or, in an older terminology,
446:is torsion-free (if the ring
1098:, or more generally for any
875:. Then one can consider the
855:is a commutative domain and
495:The torsion elements of the
91:This terminology applies to
1612:Encyclopedia of Mathematics
1473:The torsion elements of an
1057:which is a module over the
554:, which has infinite order.
402:torsion (or periodic) group
114:that are noncommutative, a
1722:
1538:Torsion-free abelian group
946:is a field, a module over
650:torsion-free abelian group
597:is an integral domain and
29:
18:Torsion (abstract algebra)
1636:10.1007/978-0-387-48899-8
1518:Annihilator (ring theory)
27:Zero divisors in a module
1528:Rank of an abelian group
1523:Localization of a module
847:Torsion and localization
823:is free. This corollary
462:is torsion-free and any
454:of non-zero-divisors of
286:be a module over a ring
118:is an element of finite
32:Torsion (disambiguation)
1665:Advanced Linear Algebra
1596:Infinite abelian groups
983:, then we may consider
954:of abelian groups from
688:Cayley–Hamilton theorem
640:, +) while the groups (
391:denotes the product of
1470:
1431:
1366:
1272:
1086:-torsion submodule of
1048:
975:). More generally, if
925:
782:
713:principal ideal domain
458:). In particular, any
399:. A group is called a
264:-modules. Since right
236:is not commutative, T(
228:, sometimes denoted T(
1581:", Birkhauser 1985,
1468:
1455:right denominator set
1432:
1367:
1273:
1100:right denominator set
1049:
926:
783:
526:by factoring out its
1696:Abelian group theory
1491:division polynomials
1376:
1286:
1216:
1211:short exact sequence
1002:
940:extension of scalars
886:
746:
565:, consisting of the
497:multiplicative group
282:More generally, let
256:) is a submodule of
30:For other uses, see
1706:Homological algebra
1607:"Torsion submodule"
1508:Arithmetic dynamics
1393:
1309:
1123:homological algebra
837:ring of polynomials
803:(depending only on
721:finitely generated
711:is a (commutative)
499:of a field are its
167:) that annihilates
65:of a module is the
49:is an element of a
1603:Michiel Hazewinkel
1533:Ray–Singer torsion
1471:
1427:
1379:
1362:
1295:
1268:
1149:yield a family of
1044:
921:
869:field of fractions
799:-module of finite
778:
668:finite-dimensional
603:field of fractions
557:The abelian group
490:Burnside's problem
486:finitely generated
460:free abelian group
409:torsion-free group
387:of the group, and
266:Noetherian domains
122:. Contrary to the
41:, specifically in
1678:978-0-387-72828-5
1645:978-0-387-98850-4
1461:Abelian varieties
518:) of 2Ă—2 integer
252:if and only if T(
222:torsion submodule
63:torsion submodule
16:(Redirected from
1713:
1681:
1656:
1619:
1592:Irving Kaplansky
1565:
1559:
1503:Analytic torsion
1444:
1436:
1434:
1433:
1428:
1420:
1415:
1414:
1392:
1387:
1371:
1369:
1368:
1363:
1361:
1360:
1336:
1331:
1330:
1308:
1303:
1277:
1275:
1274:
1269:
1258:
1253:
1252:
1240:
1239:
1053:
1051:
1050:
1045:
1040:
1039:
1030:
1029:
1014:
1013:
930:
928:
927:
922:
914:
913:
898:
897:
787:
785:
784:
779:
765:
622:torsion subgroup
567:rational numbers
411:
410:
385:identity element
340:
188:commutative ring
181:
55:non-zero-divisor
21:
1721:
1720:
1716:
1715:
1714:
1712:
1711:
1710:
1686:
1685:
1679:
1659:
1646:
1622:
1601:
1574:
1569:
1568:
1560:
1556:
1551:
1499:
1487:elliptic curves
1483:division points
1475:abelian variety
1463:
1442:
1406:
1374:
1373:
1352:
1322:
1284:
1283:
1244:
1231:
1214:
1213:
1208:
1196:
1183:
1172:-torsion of an
1159:
1119:
1081:
1068:
1031:
1021:
1005:
1000:
999:
966:
905:
889:
884:
883:
849:
744:
743:
705:
674:over the field
661:linear operator
546: ·
534:to the element
427:
408:
407:
365:torsion element
331:
184:integral domain
172:
158:regular element
154:torsion element
132:
116:torsion element
110:In the case of
47:torsion element
35:
28:
23:
22:
15:
12:
11:
5:
1719:
1717:
1709:
1708:
1703:
1698:
1688:
1687:
1684:
1683:
1677:
1661:Roman, Stephen
1657:
1644:
1624:Lam, Tsit Yuen
1620:
1599:
1589:
1573:
1570:
1567:
1566:
1553:
1552:
1550:
1547:
1546:
1545:
1540:
1535:
1530:
1525:
1520:
1515:
1510:
1505:
1498:
1495:
1479:torsion points
1462:
1459:
1426:
1423:
1419:
1413:
1409:
1405:
1402:
1399:
1396:
1391:
1386:
1382:
1359:
1355:
1351:
1348:
1345:
1342:
1339:
1335:
1329:
1325:
1321:
1318:
1315:
1312:
1307:
1302:
1298:
1294:
1291:
1267:
1264:
1261:
1257:
1251:
1247:
1243:
1238:
1234:
1230:
1227:
1224:
1221:
1206:
1203:exact sequence
1192:
1181:
1154:
1118:
1115:
1077:
1064:
1055:
1054:
1043:
1038:
1034:
1028:
1024:
1020:
1017:
1012:
1008:
962:
934:obtained from
932:
931:
920:
917:
912:
908:
904:
901:
896:
892:
848:
845:
841:direct summand
789:
788:
777:
774:
771:
768:
764:
760:
757:
754:
751:
704:
701:
700:
699:
657:
618:
581:over the ring
555:
504:
501:roots of unity
493:
478:
438:over any ring
426:
423:
260:for all right
204:torsion module
131:
128:
93:abelian groups
75:torsion module
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1718:
1707:
1704:
1702:
1701:Module theory
1699:
1697:
1694:
1693:
1691:
1680:
1674:
1670:
1666:
1662:
1658:
1655:
1651:
1647:
1641:
1637:
1633:
1629:
1625:
1621:
1618:
1614:
1613:
1608:
1604:
1600:
1597:
1593:
1590:
1588:
1587:0-8176-3065-1
1584:
1580:
1577:Ernst Kunz, "
1576:
1575:
1571:
1563:
1558:
1555:
1548:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1524:
1521:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1500:
1496:
1494:
1492:
1488:
1484:
1480:
1476:
1467:
1460:
1458:
1456:
1452:
1448:
1445:denoting the
1441:. The symbol
1440:
1421:
1417:
1411:
1407:
1403:
1400:
1394:
1389:
1384:
1380:
1357:
1353:
1346:
1337:
1333:
1327:
1323:
1319:
1316:
1310:
1305:
1300:
1296:
1289:
1281:
1265:
1259:
1255:
1249:
1245:
1236:
1232:
1225:
1219:
1212:
1204:
1200:
1195:
1191:
1187:
1179:
1175:
1171:
1167:
1163:
1157:
1152:
1148:
1144:
1141: =
1140:
1136:
1132:
1128:
1124:
1116:
1114:
1112:
1108:
1104:
1101:
1097:
1096:Ore condition
1093:
1089:
1085:
1080:
1076:
1072:
1067:
1063:
1060:
1041:
1036:
1032:
1026:
1022:
1018:
1015:
1010:
1006:
998:
997:
996:
994:
990:
986:
982:
978:
974:
970:
965:
961:
957:
953:
949:
945:
941:
937:
918:
915:
910:
906:
902:
899:
894:
890:
882:
881:
880:
878:
874:
870:
866:
863:-module. Let
862:
858:
854:
846:
844:
842:
838:
834:
831: =
830:
826:
822:
818:
814:
810:
806:
802:
798:
794:
775:
769:
758:
755:
752:
749:
742:
741:
740:
738:
734:
730:
726:
724:
718:
714:
710:
707:Suppose that
702:
697:
694:is a torsion
693:
689:
685:
681:
678:. If we view
677:
673:
670:vector space
669:
665:
662:
658:
655:
654:pure subgroup
651:
647:
643:
639:
635:
631:
627:
623:
619:
616:
613:is a torsion
612:
608:
604:
600:
596:
592:
588:
585: =
584:
580:
576:
572:
568:
564:
560:
556:
553:
549:
545:
541:
537:
533:
529:
525:
521:
517:
513:
509:
508:modular group
505:
502:
498:
494:
491:
487:
483:
479:
476:
472:
469:
465:
461:
457:
453:
449:
445:
441:
437:
433:
429:
428:
424:
422:
420:
416:
415:abelian group
412:
404:
403:
398:
394:
390:
386:
382:
378:
374:
370:
366:
362:
359:
355:
350:
348:
344:
338:
334:
329:
325:
321:
317:
313:
310:is called an
309:
305:
302:. An element
301:
297:
293:
289:
285:
280:
278:
275:
271:
267:
263:
259:
255:
251:
247:
243:
239:
235:
231:
227:
223:
220:, called the
219:
215:
211:
210:
205:
201:
197:
192:
189:
185:
179:
175:
170:
166:
162:
159:
155:
151:
148:
144:
141:
137:
129:
127:
125:
121:
117:
113:
108:
106:
102:
98:
94:
89:
87:
82:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
1664:
1627:
1610:
1564:, p. 115, §4
1557:
1482:
1478:
1472:
1450:
1438:
1372:, and hence
1279:
1198:
1193:
1189:
1185:
1177:
1173:
1169:
1165:
1161:
1155:
1153:-modules Tor
1150:
1147:Tor functors
1142:
1138:
1134:
1130:
1126:
1120:
1110:
1106:
1102:
1091:
1087:
1083:
1078:
1074:
1070:
1065:
1061:
1059:localization
1056:
992:
988:
985:localization
980:
976:
972:
963:
959:
955:
952:homomorphism
947:
943:
935:
933:
876:
872:
871:of the ring
864:
860:
856:
852:
851:Assume that
850:
832:
828:
824:
820:
812:
808:
804:
796:
792:
790:
732:
722:
716:
708:
706:
695:
691:
683:
679:
675:
671:
666:acting on a
663:
645:
641:
637:
633:
629:
625:
614:
610:
606:
598:
594:
586:
582:
578:
574:
570:
562:
558:
551:
547:
543:
539:
535:
515:
511:
482:finite group
474:
470:
464:vector space
455:
451:
447:
443:
439:
431:
418:
406:
400:
396:
392:
388:
383:denotes the
380:
376:
372:
368:
364:
363:is called a
360:
353:
351:
346:
342:
336:
332:
327:
326:annihilates
323:
319:
315:
311:
307:
303:
299:
291:
287:
283:
281:
269:
261:
257:
253:
245:
237:
233:
229:
225:
221:
217:
213:
209:torsion-free
207:
203:
202:is called a
199:
198:over a ring
195:
193:
177:
173:
168:
165:zero divisor
160:
153:
152:is called a
149:
142:
135:
133:
115:
109:
90:
83:
79:torsion-free
74:
62:
46:
36:
1513:Flat module
737:isomorphism
727:. Then the
659:Consider a
591:polynomials
524:determinant
436:free module
352:An element
272:is a right
248:is a right
134:An element
124:commutative
71:commutative
43:ring theory
39:mathematics
1690:Categories
1562:Roman 2008
1549:References
1105:and right
967:, and the
795:is a free
644:, +) and (
522:with unit
395:copies of
371:such that
322:such that
274:Noetherian
130:Definition
1617:EMS Press
1605:(2001) ,
1395:
1350:→
1344:→
1311:
1293:→
1263:→
1242:→
1229:→
1223:→
1201:) by the
1023:⊗
907:⊗
817:corollary
759:⊕
753:≃
632:, +) is (
532:conjugate
194:A module
67:submodule
1663:(2008),
1626:(2007),
1497:See also
1447:functors
1176:-module
1109:-module
991:-module
942:. Since
879:-module
825:does not
807:) and T(
698:-module.
617:-module.
520:matrices
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