701:
is reflexive, it is torsionless, thus is a submodule of a finitely generated projective module and hence is projective (semi-hereditary condition). Conversely, over a
Dedekind domain, a finitely generated torsion-free module is projective and a projective module is reflexive (the existence of a
319:
155:
355:, and for some rings there are also infinitely generated free modules that are reflexive. For instance, the direct sum of countably many copies of the integers is a reflexive module over the integers, see for instance.
182:
518:
74:
786:
761:
674:
314:{\displaystyle M\to M^{\ast \ast }=\operatorname {Hom} _{R}(M^{\ast },R),\quad m\mapsto (f\mapsto f(m)),m\in M,f\in M^{\ast },}
415:
352:
812:
376:
28:
490:
546:
380:
17:
680:
35:
782:
757:
670:
361:
745:
662:
632:
403:
24:
796:
792:
778:
753:
637:
476:
465:
407:
46:
666:
806:
770:
684:
582:
571:
529:
341:
173:
703:
345:
161:
150:{\displaystyle f\in M^{\ast }=\operatorname {Hom} _{R}(M,R),\quad f(m)\neq 0.}
468:, a finitely generated module is reflexive if and only if it is torsion-free.
325:
585:
and satisfies any of the four conditions that are known to be equivalent:
172:
A module is torsionless if and only if the canonical map into its double
360:
A submodule of a torsionless module is torsionless. In particular, any
453:-module arising in this way is torsionless (similarly, any right
372:
is a torsionless left module, and similarly for the right ideals.
777:, Graduate Texts in Mathematics No. 189, Berlin, New York:
619:(The mixture of left/right adjectives in the statement is
332:. For this reason, torsionless modules are also known as
545:
Stephen Chase proved the following characterization of
493:
328:. If this map is bijective then the module is called
185:
77:
661:. North-Holland Mathematical Library. Vol. 65.
512:
313:
149:
8:
718:, p. Ch. VII, § 4, n. 2. Proposition 8.
659:Almost Free Modules - Set-theoretic Methods
556:, the following conditions are equivalent:
483:a reflexive finitely generated module over
56:is torsionless if each non-zero element of
501:
492:
302:
228:
212:
196:
184:
101:
88:
76:
715:
549:in connection with torsionless modules:
649:
348:of torsionless modules is torsionless.
7:
657:Eklof, P. C.; Mekler, A. H. (2002).
449:-module. It turns out that any left
383:, but the converse is not true, as
351:A free module is reflexive if it is
727:
541:Relation with semihereditary rings
368:is torsionless; any left ideal of
344:is torsionless. More generally, a
14:
457:-module that is a dual of a left
45:if it can be embedded into some
246:
128:
513:{\displaystyle M\otimes _{R}S}
375:Any torsionless module over a
274:
271:
265:
259:
253:
250:
240:
221:
189:
160:This notion was introduced by
138:
132:
122:
110:
60:has non-zero image under some
1:
775:Lectures on modules and rings
667:10.1016/s0924-6509(02)x8001-5
602:Submodules of all right flat
609:Submodules of all left flat
520:is a reflexive module over
829:
445:has a structure of a left
15:
418:torsion-free module then
461:-module is torsionless).
16:Not to be confused with
563:is left semihereditary.
441:-module, then its dual
168:Properties and examples
566:All torsionless right
514:
315:
151:
515:
422:can be embedded into
316:
152:
588:All right ideals of
547:semihereditary rings
491:
183:
75:
750:Commutative algebra
595:All left ideals of
381:torsion-free module
64:-linear functional
18:Torsion-free module
613:-modules are flat.
606:-modules are flat.
510:
416:finitely generated
387:is a torsion-free
353:finitely generated
311:
147:
788:978-0-387-98428-5
746:Bourbaki, Nicolas
362:projective module
820:
799:
766:
752:(2nd ed.),
731:
725:
719:
713:
707:
695:
689:
688:
654:
519:
517:
516:
511:
506:
505:
404:commutative ring
391:-module that is
320:
318:
317:
312:
307:
306:
233:
232:
217:
216:
204:
203:
156:
154:
153:
148:
106:
105:
93:
92:
52:. Equivalently,
25:abstract algebra
828:
827:
823:
822:
821:
819:
818:
817:
803:
802:
789:
779:Springer-Verlag
769:
764:
754:Springer Verlag
744:
743:Chapter VII of
740:
735:
734:
726:
722:
714:
710:
696:
692:
677:
656:
655:
651:
646:
638:reflexive sheaf
629:
543:
497:
489:
488:
477:Noetherian ring
466:Dedekind domain
430:is torsionless.
408:integral domain
298:
224:
208:
192:
181:
180:
170:
97:
84:
73:
72:
21:
12:
11:
5:
826:
824:
816:
815:
805:
804:
801:
800:
787:
771:Lam, Tsit Yuen
767:
762:
739:
736:
733:
732:
720:
708:
690:
675:
648:
647:
645:
642:
641:
640:
635:
628:
625:
617:
616:
615:
614:
607:
600:
593:
575:
564:
542:
539:
538:
537:
509:
504:
500:
496:
469:
462:
431:
396:
373:
357:
356:
349:
334:semi-reflexive
322:
321:
310:
305:
301:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
261:
258:
255:
252:
249:
245:
242:
239:
236:
231:
227:
223:
220:
215:
211:
207:
202:
199:
195:
191:
188:
169:
166:
158:
157:
146:
143:
140:
137:
134:
131:
127:
124:
121:
118:
115:
112:
109:
104:
100:
96:
91:
87:
83:
80:
47:direct product
13:
10:
9:
6:
4:
3:
2:
825:
814:
813:Module theory
811:
810:
808:
798:
794:
790:
784:
780:
776:
772:
768:
765:
763:3-540-64239-0
759:
755:
751:
747:
742:
741:
737:
729:
724:
721:
717:
716:Bourbaki 1998
712:
709:
705:
700:
694:
691:
686:
682:
678:
676:9780444504920
672:
668:
664:
660:
653:
650:
643:
639:
636:
634:
633:Prüfer domain
631:
630:
626:
624:
622:
612:
608:
605:
601:
598:
594:
591:
587:
586:
584:
580:
576:
573:
570:-modules are
569:
565:
562:
559:
558:
557:
555:
552:For any ring
550:
548:
540:
535:
531:
527:
523:
507:
502:
498:
494:
486:
482:
478:
474:
470:
467:
463:
460:
456:
452:
448:
444:
440:
436:
433:Suppose that
432:
429:
425:
421:
417:
413:
409:
405:
401:
397:
394:
390:
386:
382:
378:
374:
371:
367:
363:
359:
358:
354:
350:
347:
343:
339:
338:
337:
335:
331:
327:
308:
303:
299:
295:
292:
289:
286:
283:
280:
277:
268:
262:
256:
247:
243:
237:
234:
229:
225:
218:
213:
209:
205:
200:
197:
193:
186:
179:
178:
177:
175:
167:
165:
163:
144:
141:
135:
129:
125:
119:
116:
113:
107:
102:
98:
94:
89:
85:
81:
78:
71:
70:
69:
67:
63:
59:
55:
51:
48:
44:
40:
37:
33:
30:
26:
19:
774:
749:
723:
711:
698:
693:
658:
652:
623:a mistake.)
620:
618:
610:
603:
596:
589:
578:
567:
560:
553:
551:
544:
533:
525:
521:
484:
480:
472:
458:
454:
450:
446:
442:
438:
434:
427:
426:, and hence
423:
419:
411:
399:
395:torsionless.
392:
388:
384:
369:
365:
333:
329:
323:
171:
159:
65:
61:
57:
53:
49:
42:
38:
31:
22:
437:is a right
406:that is an
342:free module
43:torsionless
738:References
704:dual basis
697:Proof: If
346:direct sum
162:Hyman Bass
41:is called
685:116961421
599:are flat.
592:are flat.
577:The ring
524:whenever
499:⊗
340:A unital
330:reflexive
326:injective
304:∗
296:∈
284:∈
260:↦
251:↦
230:∗
219:
201:∗
198:∗
190:→
142:≠
108:
90:∗
82:∈
807:Category
773:(1999),
748:(1998),
730:, p 146.
728:Lam 1999
627:See also
583:coherent
581:is left
797:1653294
487:. Then
464:Over a
34:over a
795:
785:
760:
683:
673:
377:domain
29:module
681:S2CID
532:over
475:be a
414:is a
402:is a
379:is a
364:over
783:ISBN
758:ISBN
671:ISBN
644:Note
572:flat
530:flat
479:and
471:Let
410:and
174:dual
36:ring
27:, a
663:doi
621:not
528:is
398:If
393:not
324:is
210:Hom
99:Hom
23:In
809::
793:MR
791:,
781:,
756:,
706:).
679:.
669:.
336:.
176:,
164:.
145:0.
68::
699:M
687:.
665::
611:R
604:R
597:R
590:R
579:R
574:.
568:R
561:R
554:R
536:.
534:R
526:S
522:S
508:S
503:R
495:M
485:R
481:M
473:R
459:R
455:R
451:R
447:R
443:N
439:R
435:N
428:M
424:R
420:M
412:M
400:R
389:Z
385:Q
370:R
366:R
309:,
300:M
293:f
290:,
287:M
281:m
278:,
275:)
272:)
269:m
266:(
263:f
257:f
254:(
248:m
244:,
241:)
238:R
235:,
226:M
222:(
214:R
206:=
194:M
187:M
139:)
136:m
133:(
130:f
126:,
123:)
120:R
117:,
114:M
111:(
103:R
95:=
86:M
79:f
66:f
62:R
58:M
54:M
50:R
39:R
32:M
20:.
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