1175:
643:, i.e. cannot be expressed as a direct sum of a pair of their proper subgroups. These examples demonstrate that torsion-free abelian group of rank greater than 1 cannot be simply built by direct sums from torsion-free abelian groups of rank 1, whose theory is well understood. Moreover, for every integer
610:
211:
1068:
512:
1145:
866:
466:
868:. Thus the sequence of ranks of indecomposable summands in a certain direct sum decomposition of a torsion-free abelian group of finite rank is very far from being an invariant of
795:
735:
667:
410:
and there is a satisfactory classification of them up to isomorphism. By contrast, there is no satisfactory classification of torsion-free abelian groups of rank 2.
696:
138:
702:
indecomposable groups. Hence even the number of indecomposable summands of a group of an even rank greater or equal than 4 is not well-defined.
995:
1089:
It is a generalization, since every abelian group is a module over the integers. It easily follows that the dimension of the product over
1239:
1211:
1192:
1316:
Thomas, Simon; Schneider, Scott (2012), "Countable Borel equivalence relations", in
Cummings, James; Schimmerling, Ernest (eds.),
1218:
225:
are zero (so that the sum is, in effect, finite), then all coefficients are zero. Any two maximal linearly independent sets in
1293:
403:
91:
1225:
605:{\displaystyle \operatorname {rank} \left(\bigoplus _{j\in J}A_{j}\right)=\sum _{j\in J}\operatorname {rank} (A_{j}),}
1343:
1258:
1207:
83:
1109:
312:
1196:
812:
1320:, London Mathematical Society Lecture Note Series, vol. 406, Cambridge University Press, pp. 25–62,
1369:
1353:, Thomas and Schneider refer to "...this failure to classify even the rank 2 groups in a satisfactory way..."
426:
631:
Abelian groups of rank greater than 1 are sources of interesting examples. For instance, for every cardinal
748:
705:
Another result about non-uniqueness of direct sum decompositions is due to A.L.S. Corner: given integers
94:
have been completely classified. However, the theory of abelian groups of higher rank is more involved.
1232:
1326:
708:
98:
86:, rank is a strong invariant and every such group is determined up to isomorphism by its rank and
1185:
1321:
245:
276:). A group is called torsion-free if it has no non-trivial torsion elements. The factor-group
1350:
646:
640:
414:
300:
261:
124:
47:
1303:
672:
206:{\displaystyle \sum _{\alpha }n_{\alpha }a_{\alpha }=0,\quad n_{\alpha }\in \mathbb {Z} ,}
8:
1093:
is the cardinality of maximal linearly independent subset, since for any torsion element
619:
132:) if the only linear combination of these elements that is equal to zero is trivial: if
55:
1339:
1289:
253:
67:
1331:
1299:
265:
87:
75:
1335:
1158:
966:
304:
984:
980:
392:
1363:
36:
1083:
249:
71:
487:
230:
43:
20:
1063:{\displaystyle \operatorname {rank} (M)=\dim _{R_{0}}M\otimes _{R}R_{0}}
698:
that is simultaneously a sum of two indecomposable groups, and a sum of
1281:
500:
357:
264:
is finite. The set of all torsion elements is a subgroup, called the
252:. The main difference with the case of vector space is a presence of
1174:
920:
For abelian groups of infinite rank, there is an example of a group
299:
The notion of rank with analogous properties can be defined for
372:
as an abelian subgroup. In particular, any intermediate group
875:
Other surprising examples include torsion-free rank 2 groups
307:, the case of abelian groups corresponding to modules over
1082:
is a field, and thus any module (or, to be more specific,
474:
is a short exact sequence of abelian groups then rk
97:
The term rank has a different meaning in the context of
1112:
998:
961:
The notion of rank can be generalized for any module
815:
751:
711:
675:
649:
515:
429:
141:
16:
Number of elements in a subset of a commutative group
1199:. Unsourced material may be challenged and removed.
1288:(Third ed.), Reading, Mass.: Addison-Wesley,
1139:
1062:
860:
789:
729:
690:
661:
604:
460:
205:
288:) is the unique maximal torsion-free quotient of
244:The rank of an abelian group is analogous to the
1361:
669:, there is a torsion-free abelian group of rank
635:there exist torsion-free abelian groups of rank
1315:
494:and the corresponding fact for vector spaces.
1140:{\displaystyle x\otimes _{\mathbf {Z} }q=0.}
737:, there exists a torsion-free abelian group
618:where the sum in the right hand side uses
457:
1325:
1259:Learn how and when to remove this message
861:{\displaystyle r_{1},r_{2},\ldots ,r_{k}}
391:Abelian groups of rank 0 are exactly the
216:where all but finitely many coefficients
196:
626:
292:and its rank coincides with the rank of
461:{\displaystyle 0\to A\to B\to C\to 0\;}
344:is torsion-free then the canonical map
1362:
313:finitely generated module#Generic rank
790:{\displaystyle n=r_{1}+\cdots +r_{k}}
404:Torsion-free abelian groups of rank 1
92:Torsion-free abelian groups of rank 1
1280:
1197:adding citations to reliable sources
1168:
328:coincides with the dimension of the
54:determines the size of the largest
13:
809:indecomposable subgroups of ranks
14:
1381:
1318:Appalachian Set Theory: 2006-2012
956:
256:. An element of an abelian group
84:finitely generated abelian groups
1173:
1122:
948:Every nonzero direct summand of
499:Rank is additive over arbitrary
402:of rational numbers has rank 1.
260:is classified as torsion if its
1184:needs additional citations for
945:and a single other element; and
181:
1309:
1273:
1011:
1005:
987:of the module with the field:
596:
583:
451:
445:
439:
433:
1:
1164:
730:{\displaystyle n\geq k\geq 1}
406:are realized as subgroups of
324:The rank of an abelian group
318:
104:
1336:10.1017/CBO9781139208574.003
801:natural summands, the group
745:such that for any partition
364:is the minimum dimension of
7:
1152:
10:
1386:
1208:"Rank of an abelian group"
368:-vector space containing
99:elementary abelian groups
972:, as the dimension over
486:. This follows from the
662:{\displaystyle n\geq 3}
393:periodic abelian groups
1141:
1075:It makes sense, since
1064:
862:
791:
731:
692:
663:
606:
462:
413:Rank is additive over
233:, which is called the
207:
118:} of an abelian group
70:then it embeds into a
1142:
1065:
863:
805:is the direct sum of
792:
732:
693:
664:
627:Groups of higher rank
607:
482: + rk
463:
415:short exact sequences
208:
1370:Abelian group theory
1193:improve this article
1110:
996:
813:
749:
709:
691:{\displaystyle 2n-2}
673:
647:
513:
427:
139:
125:linearly independent
50:subset. The rank of
48:linearly independent
1086:) over it is free.
620:cardinal arithmetic
1137:
1060:
935:is indecomposable;
858:
787:
727:
688:
659:
602:
576:
542:
458:
203:
151:
78:of dimension rank
56:free abelian group
1295:978-0-201-55540-0
1269:
1268:
1261:
1243:
1097:and any rational
905:is isomorphic to
561:
527:
142:
33:torsion-free rank
1377:
1354:
1348:
1329:
1313:
1307:
1306:
1277:
1264:
1257:
1253:
1250:
1244:
1242:
1201:
1177:
1169:
1146:
1144:
1143:
1138:
1127:
1126:
1125:
1069:
1067:
1066:
1061:
1059:
1058:
1049:
1048:
1033:
1032:
1031:
1030:
952:is decomposable.
941:is generated by
913:is divisible by
867:
865:
864:
859:
857:
856:
838:
837:
825:
824:
796:
794:
793:
788:
786:
785:
767:
766:
736:
734:
733:
728:
697:
695:
694:
689:
668:
666:
665:
660:
611:
609:
608:
603:
595:
594:
575:
557:
553:
552:
551:
541:
467:
465:
464:
459:
360:and the rank of
311:. For this, see
266:torsion subgroup
212:
210:
209:
204:
199:
191:
190:
171:
170:
161:
160:
150:
88:torsion subgroup
76:rational numbers
1385:
1384:
1380:
1379:
1378:
1376:
1375:
1374:
1360:
1359:
1358:
1357:
1346:
1327:10.1.1.648.3113
1314:
1310:
1296:
1278:
1274:
1265:
1254:
1248:
1245:
1202:
1200:
1190:
1178:
1167:
1159:Rank of a group
1155:
1121:
1120:
1116:
1111:
1108:
1107:
1081:
1054:
1050:
1044:
1040:
1026:
1022:
1021:
1017:
997:
994:
993:
978:
967:integral domain
959:
924:and a subgroup
909:if and only if
900:
887:
852:
848:
833:
829:
820:
816:
814:
811:
810:
781:
777:
762:
758:
750:
747:
746:
710:
707:
706:
674:
671:
670:
648:
645:
644:
629:
590:
586:
565:
547:
543:
531:
526:
522:
514:
511:
510:
478:= rk
428:
425:
424:
321:
305:integral domain
224:
195:
186:
182:
166:
162:
156:
152:
146:
140:
137:
136:
117:
107:
17:
12:
11:
5:
1383:
1373:
1372:
1356:
1355:
1344:
1308:
1294:
1271:
1270:
1267:
1266:
1249:September 2008
1181:
1179:
1172:
1166:
1163:
1162:
1161:
1154:
1151:
1150:
1149:
1148:
1147:
1136:
1133:
1130:
1124:
1119:
1115:
1079:
1073:
1072:
1071:
1070:
1057:
1053:
1047:
1043:
1039:
1036:
1029:
1025:
1020:
1016:
1013:
1010:
1007:
1004:
1001:
985:tensor product
981:quotient field
976:
958:
957:Generalization
955:
954:
953:
946:
936:
892:
879:
855:
851:
847:
844:
841:
836:
832:
828:
823:
819:
784:
780:
776:
773:
770:
765:
761:
757:
754:
726:
723:
720:
717:
714:
687:
684:
681:
678:
658:
655:
652:
641:indecomposable
628:
625:
624:
623:
615:
614:
613:
612:
601:
598:
593:
589:
585:
582:
579:
574:
571:
568:
564:
560:
556:
550:
546:
540:
537:
534:
530:
525:
521:
518:
505:
504:
496:
495:
471:
470:
469:
468:
456:
453:
450:
447:
444:
441:
438:
435:
432:
419:
418:
411:
396:
389:
332:-vector space
320:
317:
229:have the same
220:
214:
213:
202:
198:
194:
189:
185:
180:
177:
174:
169:
165:
159:
155:
149:
145:
113:
106:
103:
15:
9:
6:
4:
3:
2:
1382:
1371:
1368:
1367:
1365:
1352:
1347:
1345:9781107608504
1341:
1337:
1333:
1328:
1323:
1319:
1312:
1305:
1301:
1297:
1291:
1287:
1283:
1276:
1272:
1263:
1260:
1252:
1241:
1238:
1234:
1231:
1227:
1224:
1220:
1217:
1213:
1210: –
1209:
1205:
1204:Find sources:
1198:
1194:
1188:
1187:
1182:This article
1180:
1176:
1171:
1170:
1160:
1157:
1156:
1134:
1131:
1128:
1117:
1113:
1106:
1105:
1104:
1103:
1102:
1100:
1096:
1092:
1087:
1085:
1078:
1055:
1051:
1045:
1041:
1037:
1034:
1027:
1023:
1018:
1014:
1008:
1002:
999:
992:
991:
990:
989:
988:
986:
982:
975:
971:
968:
964:
951:
947:
944:
940:
937:
934:
931:
930:
929:
927:
923:
918:
916:
912:
908:
904:
899:
895:
891:
886:
882:
878:
873:
871:
853:
849:
845:
842:
839:
834:
830:
826:
821:
817:
808:
804:
800:
782:
778:
774:
771:
768:
763:
759:
755:
752:
744:
740:
724:
721:
718:
715:
712:
703:
701:
685:
682:
679:
676:
656:
653:
650:
642:
638:
634:
621:
617:
616:
599:
591:
587:
580:
577:
572:
569:
566:
562:
558:
554:
548:
544:
538:
535:
532:
528:
523:
519:
516:
509:
508:
507:
506:
502:
498:
497:
493:
489:
485:
481:
477:
473:
472:
454:
448:
442:
436:
430:
423:
422:
421:
420:
416:
412:
409:
405:
401:
397:
394:
390:
387:
383:
379:
375:
371:
367:
363:
359:
355:
351:
347:
343:
339:
335:
331:
327:
323:
322:
316:
314:
310:
306:
302:
297:
295:
291:
287:
283:
279:
275:
271:
267:
263:
259:
255:
251:
247:
242:
240:
236:
232:
228:
223:
219:
200:
192:
187:
183:
178:
175:
172:
167:
163:
157:
153:
147:
143:
135:
134:
133:
131:
127:
126:
121:
116:
112:
102:
100:
95:
93:
89:
85:
81:
77:
73:
69:
65:
61:
58:contained in
57:
53:
49:
46:of a maximal
45:
41:
38:
37:abelian group
34:
30:
26:
22:
1317:
1311:
1285:
1275:
1255:
1246:
1236:
1229:
1222:
1215:
1203:
1191:Please help
1186:verification
1183:
1098:
1094:
1090:
1088:
1084:vector space
1076:
1074:
973:
969:
962:
960:
949:
942:
938:
932:
925:
921:
919:
914:
910:
906:
902:
897:
893:
889:
884:
880:
876:
874:
869:
806:
802:
798:
742:
738:
704:
699:
636:
632:
630:
491:
483:
479:
475:
407:
399:
385:
381:
377:
373:
369:
365:
361:
353:
349:
345:
341:
337:
333:
329:
325:
308:
298:
293:
289:
285:
281:
277:
273:
269:
268:and denoted
257:
250:vector space
243:
238:
234:
226:
221:
217:
215:
129:
123:
119:
114:
110:
108:
96:
79:
72:vector space
68:torsion-free
63:
59:
51:
39:
32:
28:
24:
18:
1282:Lang, Serge
1279:Page 46 of
928:such that
501:direct sums
231:cardinality
44:cardinality
29:Prüfer rank
21:mathematics
1351:p. 46
1304:0848.13001
1219:newspapers
1165:References
901:such that
398:The group
319:Properties
109:A subset {
105:Definition
1322:CiteSeerX
1118:⊗
1042:⊗
1035:
1003:
983:, of the
843:…
772:⋯
722:≥
716:≥
683:−
654:≥
639:that are
581:
570:∈
563:∑
536:∈
529:⨁
520:
452:→
446:→
440:→
434:→
384:has rank
358:injective
303:over any
246:dimension
193:∈
188:α
168:α
158:α
148:α
144:∑
74:over the
1364:Category
1284:(1993),
1153:See also
965:over an
741:of rank
488:flatness
1286:Algebra
1233:scholar
301:modules
254:torsion
42:is the
1342:
1324:
1302:
1292:
1235:
1228:
1221:
1214:
1206:
979:, the
128:(over
82:. For
35:of an
23:, the
1349:. On
1240:JSTOR
1226:books
797:into
380:<
376:<
340:. If
262:order
248:of a
62:. If
31:, or
1340:ISBN
1290:ISBN
1212:news
1000:rank
888:and
578:rank
517:rank
417:: if
235:rank
25:rank
1332:doi
1300:Zbl
1195:by
1019:dim
490:of
356:is
237:of
122:is
66:is
19:In
1366::
1338:,
1330:,
1298:,
1135:0.
1101:,
917:.
872:.
352:⊗
348:→
336:⊗
315:.
296:.
241:.
101:.
90:.
27:,
1334::
1262:)
1256:(
1251:)
1247:(
1237:·
1230:·
1223:·
1216:·
1189:.
1132:=
1129:q
1123:Z
1114:x
1099:q
1095:x
1091:Q
1080:0
1077:R
1056:0
1052:R
1046:R
1038:M
1028:0
1024:R
1015:=
1012:)
1009:M
1006:(
977:0
974:R
970:R
963:M
950:G
943:G
939:K
933:K
926:G
922:K
915:m
911:n
907:B
903:A
898:m
896:,
894:n
890:B
885:m
883:,
881:n
877:A
870:A
854:k
850:r
846:,
840:,
835:2
831:r
827:,
822:1
818:r
807:k
803:A
799:k
783:k
779:r
775:+
769:+
764:1
760:r
756:=
753:n
743:n
739:A
725:1
719:k
713:n
700:n
686:2
680:n
677:2
657:3
651:n
637:d
633:d
622:.
600:,
597:)
592:j
588:A
584:(
573:J
567:j
559:=
555:)
549:j
545:A
539:J
533:j
524:(
503::
492:Q
484:C
480:A
476:B
455:0
449:C
443:B
437:A
431:0
408:Q
400:Q
395:.
388:.
386:n
382:Q
378:A
374:Z
370:A
366:Q
362:A
354:Q
350:A
346:A
342:A
338:Q
334:A
330:Q
326:A
309:Z
294:A
290:A
286:A
284:(
282:T
280:/
278:A
274:A
272:(
270:T
258:A
239:A
227:A
222:α
218:n
201:,
197:Z
184:n
179:,
176:0
173:=
164:a
154:n
130:Z
120:A
115:α
111:a
80:A
64:A
60:A
52:A
40:A
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.