1455:
179:
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
1359:
717:
1424:
775:
1265:
1041:
918:
1083:
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
828:
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
1274:
481:, conversely, asks whether a finitely generated periodic group must be finite. The answer is "no" in general, even if the period is fixed.
1438:
reflects this relation with the algebraic torsion. This same result holds for non-commutative rings as well as long as the set
675:-module in the natural way, then (as a result of many things, either simply by finite-dimensionality or as a consequence of the
96:(in fact, this is the origin of the terminology, which was introduced for abelian groups before being generalized to modules).
1665:
1632:
1364:
734:
1575:
1204:
990:
284:
1684:
1657:
1605:
1531:
146:
1694:
874:
1600:
676:
1595:
1526:
638:
520:
1506:
709:
1516:
1511:
973:
656:
474:
58:
formed by the torsion elements (in cases when this is indeed a submodule, such as when the ring is
20:
1443:
1088:
1689:
1584:
1567:
1521:
1047:
701:
516:
478:
1642:
1479:
1199:
928:
508:
485:
265:
108:
74:
39:
8:
1496:
1191:
1111:
825:
456:
346:
197:
100:
92:"). This is allowed by the fact that the abelian groups are the modules over the ring of
85:
67:
1591:
940:
857:
591:
582:
in one variable is pure torsion. Both these examples can be generalized as follows: if
448:
135:
47:
93:
1661:
1628:
1571:
254:
558:
modulo 1, is periodic, i.e. every element has finite order. Analogously, the module
1620:
1580:
1491:
957:
610:
373:
176:
128:
59:
43:
1638:
1463:
649:
555:
262:
172:
1475:
829:
77:, that is, when the regular elements of the ring are all its nonzero elements.
1624:
718:
structure theorem for finitely generated modules over a principal ideal domain
1678:
1612:
1084:
642:
496:
489:
403:
390:
238:
112:
81:
1649:
470:
452:
153:
1354:{\displaystyle 0\to \operatorname {Tor} _{1}^{R}(M,R_{S}/R)\to M\to M_{S}}
356:
of the group if it has finite order, i.e., if there is a positive integer
1501:
1454:
1135:
789:
725:
512:
424:
205:
is commutative then the set of all torsion elements forms a submodule of
31:
27:
1619:, Problem Books in Mathematics, New York: Springer, pp. xviii+412,
939:
is a vector space, possibly infinite-dimensional. There is a canonical
579:
531:. In this case, torsion elements do not form a subgroup, for example,
1458:
The 4-torsion subgroup of an elliptic curve over the complex numbers.
805:
55:
439:
is not a domain then torsion is considered with respect to the set
410:
of integers, and in this case the two notions of torsion coincide.
89:
1435:
66:
is a module consisting entirely of torsion elements. A module is
115:
case, the torsion elements do not form a subgroup, in general.
641:
by a subgroup is torsion-free exactly when the subgroup is a
519:, any nontrivial torsion element either has order two and is
152:
of the ring (an element that is neither a left nor a right
1568:
Introduction to
Commutative algebra and algebraic geometry
960:
of this homomorphism is precisely the torsion submodule T(
73:
This terminology is more commonly used for modules over a
402:
if its only torsion element is the identity element. Any
431:. Then it follows immediately from the definitions that
1419:{\displaystyle \operatorname {Tor} _{1}^{R}(M,R_{S}/R)}
1367:
1277:
1207:
993:
877:
737:
816:
hold for more general commutative domains, even for
691:
527:or has order three and is conjugate to the element
1590:
1418:
1353:
1259:
1110:The concept of torsion plays an important role in
1035:
912:
808:, any finitely generated torsion-free module over
769:
1105:
229:) may or may not be a submodule. It is shown in (
201:if zero is the only torsion element. If the ring
70:if its only torsion element is the zero element.
1676:
968:is a multiplicatively closed subset of the ring
770:{\displaystyle M\simeq F\oplus \mathrm {T} (M),}
394:if all its elements are torsion elements, and a
1260:{\displaystyle 0\to R\to R_{S}\to R_{S}/R\to 0}
195:if all its elements are torsion elements, and
462:is torsion-free when viewed as a module over
303:-torsion element if there exists an element
84:(with "module" and "submodule" replaced by "
835:
720:gives a detailed description of the module
1122:are two modules over a commutative domain
637:, +) are torsion-free. The quotient of a
1660:(Third ed.), Springer, p. 446,
1426:is the kernel of the localisation map of
1036:{\displaystyle M_{S}=M\otimes _{R}R_{S},}
42:that yields zero when multiplied by some
1453:
406:may be viewed as a module over the ring
334:the set of regular elements of the ring
1126:(for example, two abelian groups, when
1677:
1648:
1550:
913:{\displaystyle M_{Q}=M\otimes _{R}Q,}
1449:
1611:
257:are Ore, this covers the case when
230:
13:
1271:-modules yields an exact sequence
751:
338:and recover the definition above.
268:(which might not be commutative).
14:
1706:
1478:they may be computed in terms of
473:(abelian or not) is periodic and
1587:", University of Michigan, 1954.
1169:is canonically isomorphic to Tor
1079:. Thus the torsion submodule of
1071:, whose kernel is precisely the
1058:. There is a canonical map from
728:. In particular, it claims that
692:Case of a principal ideal domain
469:By contrast with example 1, any
330:In particular, one can take for
145:of the module if there exists a
1617:Exercises in modules and rings
1544:
1413:
1386:
1338:
1332:
1329:
1302:
1281:
1251:
1230:
1217:
1211:
1106:Torsion in homological algebra
800:) is the torsion submodule of
761:
755:
503:obtained from the group SL(2,
285:multiplicatively closed subset
1:
1658:Graduate Texts in Mathematics
1537:
1532:Universal coefficient theorem
1470:or, in an older terminology,
435:is torsion-free (if the ring
118:
1087:, or more generally for any
864:. Then one can consider the
844:is a commutative domain and
484:The torsion elements of the
80:This terminology applies to
7:
1601:Encyclopedia of Mathematics
1485:
1462:The torsion elements of an
1046:which is a module over the
543:, which has infinite order.
413:
391:torsion (or periodic) group
103:that are noncommutative, a
10:
1711:
1560:
1527:Torsion-free abelian group
935:is a field, a module over
639:torsion-free abelian group
586:is an integral domain and
18:
1625:10.1007/978-0-387-48899-8
1507:Annihilator (ring theory)
16:Zero divisors in a module
1517:Rank of an abelian group
1512:Localization of a module
836:Torsion and localization
812:is free. This corollary
451:is torsion-free and any
443:of non-zero-divisors of
275:be a module over a ring
107:is an element of finite
21:Torsion (disambiguation)
1654:Advanced Linear Algebra
1585:Infinite abelian groups
972:, then we may consider
943:of abelian groups from
677:Cayley–Hamilton theorem
629:, +) while the groups (
380:denotes the product of
1459:
1420:
1355:
1261:
1075:-torsion submodule of
1037:
964:). More generally, if
914:
771:
702:principal ideal domain
447:). In particular, any
388:. A group is called a
253:-modules. Since right
225:is not commutative, T(
217:, sometimes denoted T(
1570:", Birkhauser 1985,
1457:
1444:right denominator set
1421:
1356:
1262:
1089:right denominator set
1038:
915:
772:
515:by factoring out its
1685:Abelian group theory
1480:division polynomials
1365:
1275:
1205:
1200:short exact sequence
991:
929:extension of scalars
875:
735:
554:, consisting of the
486:multiplicative group
271:More generally, let
245:) is a submodule of
19:For other uses, see
1695:Homological algebra
1596:"Torsion submodule"
1497:Arithmetic dynamics
1382:
1298:
1112:homological algebra
826:ring of polynomials
792:(depending only on
710:finitely generated
700:is a (commutative)
488:of a field are its
156:) that annihilates
54:of a module is the
38:is an element of a
1592:Michiel Hazewinkel
1522:Ray–Singer torsion
1460:
1416:
1368:
1351:
1284:
1257:
1138:yield a family of
1033:
910:
858:field of fractions
788:-module of finite
767:
657:finite-dimensional
592:field of fractions
546:The abelian group
479:Burnside's problem
475:finitely generated
449:free abelian group
398:torsion-free group
376:of the group, and
255:Noetherian domains
111:. Contrary to the
30:, specifically in
1667:978-0-387-72828-5
1634:978-0-387-98850-4
1450:Abelian varieties
507:) of 2Ă—2 integer
241:if and only if T(
211:torsion submodule
52:torsion submodule
1702:
1670:
1645:
1608:
1581:Irving Kaplansky
1554:
1548:
1492:Analytic torsion
1433:
1425:
1423:
1422:
1417:
1409:
1404:
1403:
1381:
1376:
1360:
1358:
1357:
1352:
1350:
1349:
1325:
1320:
1319:
1297:
1292:
1266:
1264:
1263:
1258:
1247:
1242:
1241:
1229:
1228:
1042:
1040:
1039:
1034:
1029:
1028:
1019:
1018:
1003:
1002:
919:
917:
916:
911:
903:
902:
887:
886:
776:
774:
773:
768:
754:
611:torsion subgroup
556:rational numbers
400:
399:
374:identity element
329:
177:commutative ring
170:
44:non-zero-divisor
1710:
1709:
1705:
1704:
1703:
1701:
1700:
1699:
1675:
1674:
1668:
1635:
1563:
1558:
1557:
1549:
1545:
1540:
1488:
1476:elliptic curves
1472:division points
1464:abelian variety
1452:
1431:
1405:
1399:
1395:
1377:
1372:
1366:
1363:
1362:
1345:
1341:
1321:
1315:
1311:
1293:
1288:
1276:
1273:
1272:
1243:
1237:
1233:
1224:
1220:
1206:
1203:
1202:
1197:
1185:
1172:
1161:-torsion of an
1148:
1108:
1070:
1057:
1024:
1020:
1014:
1010:
998:
994:
992:
989:
988:
955:
898:
894:
882:
878:
876:
873:
872:
838:
750:
736:
733:
732:
694:
663:over the field
650:linear operator
535: ·
523:to the element
416:
397:
396:
354:torsion element
320:
173:integral domain
161:
147:regular element
143:torsion element
121:
105:torsion element
99:In the case of
36:torsion element
24:
17:
12:
11:
5:
1708:
1698:
1697:
1692:
1687:
1673:
1672:
1666:
1650:Roman, Stephen
1646:
1633:
1613:Lam, Tsit Yuen
1609:
1588:
1578:
1562:
1559:
1556:
1555:
1542:
1541:
1539:
1536:
1535:
1534:
1529:
1524:
1519:
1514:
1509:
1504:
1499:
1494:
1487:
1484:
1468:torsion points
1451:
1448:
1415:
1412:
1408:
1402:
1398:
1394:
1391:
1388:
1385:
1380:
1375:
1371:
1348:
1344:
1340:
1337:
1334:
1331:
1328:
1324:
1318:
1314:
1310:
1307:
1304:
1301:
1296:
1291:
1287:
1283:
1280:
1256:
1253:
1250:
1246:
1240:
1236:
1232:
1227:
1223:
1219:
1216:
1213:
1210:
1195:
1192:exact sequence
1181:
1170:
1143:
1107:
1104:
1066:
1053:
1044:
1043:
1032:
1027:
1023:
1017:
1013:
1009:
1006:
1001:
997:
951:
923:obtained from
921:
920:
909:
906:
901:
897:
893:
890:
885:
881:
837:
834:
830:direct summand
778:
777:
766:
763:
760:
757:
753:
749:
746:
743:
740:
693:
690:
689:
688:
646:
607:
570:over the ring
544:
493:
490:roots of unity
482:
467:
427:over any ring
415:
412:
249:for all right
193:torsion module
120:
117:
82:abelian groups
64:torsion module
15:
9:
6:
4:
3:
2:
1707:
1696:
1693:
1691:
1690:Module theory
1688:
1686:
1683:
1682:
1680:
1669:
1663:
1659:
1655:
1651:
1647:
1644:
1640:
1636:
1630:
1626:
1622:
1618:
1614:
1610:
1607:
1603:
1602:
1597:
1593:
1589:
1586:
1582:
1579:
1577:
1576:0-8176-3065-1
1573:
1569:
1566:Ernst Kunz, "
1565:
1564:
1552:
1547:
1543:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1513:
1510:
1508:
1505:
1503:
1500:
1498:
1495:
1493:
1490:
1489:
1483:
1481:
1477:
1473:
1469:
1465:
1456:
1447:
1445:
1441:
1437:
1434:denoting the
1430:. The symbol
1429:
1410:
1406:
1400:
1396:
1392:
1389:
1383:
1378:
1373:
1369:
1346:
1342:
1335:
1326:
1322:
1316:
1312:
1308:
1305:
1299:
1294:
1289:
1285:
1278:
1270:
1254:
1248:
1244:
1238:
1234:
1225:
1221:
1214:
1208:
1201:
1193:
1189:
1184:
1180:
1176:
1168:
1164:
1160:
1156:
1152:
1146:
1141:
1137:
1133:
1130: =
1129:
1125:
1121:
1117:
1113:
1103:
1101:
1097:
1093:
1090:
1086:
1085:Ore condition
1082:
1078:
1074:
1069:
1065:
1061:
1056:
1052:
1049:
1030:
1025:
1021:
1015:
1011:
1007:
1004:
999:
995:
987:
986:
985:
983:
979:
975:
971:
967:
963:
959:
954:
950:
946:
942:
938:
934:
930:
926:
907:
904:
899:
895:
891:
888:
883:
879:
871:
870:
869:
867:
863:
859:
855:
852:-module. Let
851:
847:
843:
833:
831:
827:
823:
820: =
819:
815:
811:
807:
803:
799:
795:
791:
787:
783:
764:
758:
747:
744:
741:
738:
731:
730:
729:
727:
723:
719:
715:
713:
707:
703:
699:
696:Suppose that
686:
683:is a torsion
682:
678:
674:
670:
667:. If we view
666:
662:
659:vector space
658:
654:
651:
647:
644:
643:pure subgroup
640:
636:
632:
628:
624:
620:
616:
612:
608:
605:
602:is a torsion
601:
597:
593:
589:
585:
581:
577:
574: =
573:
569:
565:
561:
557:
553:
549:
545:
542:
538:
534:
530:
526:
522:
518:
514:
510:
506:
502:
498:
497:modular group
494:
491:
487:
483:
480:
476:
472:
468:
465:
461:
458:
454:
450:
446:
442:
438:
434:
430:
426:
422:
418:
417:
411:
409:
405:
404:abelian group
401:
393:
392:
387:
383:
379:
375:
371:
367:
363:
359:
355:
351:
348:
344:
339:
337:
333:
327:
323:
318:
314:
310:
306:
302:
299:is called an
298:
294:
291:. An element
290:
286:
282:
278:
274:
269:
267:
264:
260:
256:
252:
248:
244:
240:
236:
232:
228:
224:
220:
216:
212:
209:, called the
208:
204:
200:
199:
194:
190:
186:
181:
178:
174:
168:
164:
159:
155:
151:
148:
144:
140:
137:
133:
130:
126:
116:
114:
110:
106:
102:
97:
95:
91:
87:
83:
78:
76:
71:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
29:
22:
1653:
1616:
1599:
1553:, p. 115, §4
1546:
1471:
1467:
1461:
1439:
1427:
1361:, and hence
1268:
1187:
1182:
1178:
1174:
1166:
1162:
1158:
1154:
1150:
1144:
1142:-modules Tor
1139:
1136:Tor functors
1131:
1127:
1123:
1119:
1115:
1109:
1099:
1095:
1091:
1080:
1076:
1072:
1067:
1063:
1059:
1054:
1050:
1048:localization
1045:
981:
977:
974:localization
969:
965:
961:
952:
948:
944:
941:homomorphism
936:
932:
924:
922:
865:
861:
860:of the ring
853:
849:
845:
841:
840:Assume that
839:
821:
817:
813:
809:
801:
797:
793:
785:
781:
779:
721:
711:
705:
697:
695:
684:
680:
672:
668:
664:
660:
655:acting on a
652:
634:
630:
626:
622:
618:
614:
603:
599:
595:
587:
583:
575:
571:
567:
563:
559:
551:
547:
540:
536:
532:
528:
524:
504:
500:
471:finite group
463:
459:
453:vector space
444:
440:
436:
432:
428:
420:
407:
395:
389:
385:
381:
377:
372:denotes the
369:
365:
361:
357:
353:
352:is called a
349:
342:
340:
335:
331:
325:
321:
316:
315:annihilates
312:
308:
304:
300:
296:
292:
288:
280:
276:
272:
270:
258:
250:
246:
242:
234:
226:
222:
218:
214:
210:
206:
202:
198:torsion-free
196:
192:
191:is called a
188:
187:over a ring
184:
182:
166:
162:
157:
154:zero divisor
149:
142:
141:is called a
138:
131:
124:
122:
104:
98:
79:
72:
68:torsion-free
63:
51:
35:
25:
1502:Flat module
726:isomorphism
716:. Then the
648:Consider a
580:polynomials
513:determinant
425:free module
341:An element
261:is a right
237:is a right
123:An element
113:commutative
60:commutative
32:ring theory
28:mathematics
1679:Categories
1551:Roman 2008
1538:References
1094:and right
956:, and the
784:is a free
633:, +) and (
511:with unit
384:copies of
360:such that
311:such that
263:Noetherian
119:Definition
1606:EMS Press
1594:(2001) ,
1384:
1339:→
1333:→
1300:
1282:→
1252:→
1231:→
1218:→
1212:→
1190:) by the
1012:⊗
896:⊗
806:corollary
748:⊕
742:≃
621:, +) is (
521:conjugate
183:A module
56:submodule
1652:(2008),
1615:(2007),
1486:See also
1436:functors
1165:-module
1098:-module
980:-module
931:. Since
868:-module
814:does not
796:) and T(
687:-module.
606:-module.
509:matrices
414:Examples
368:, where
319:, i.e.,
239:Ore ring
231:Lam 2007
160:, i.e.,
94:integers
90:subgroup
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