193:
627:
668:
104:
687:
362:
661:
588:
459:
148:
17:
697:
580:
654:
243:
204:
469:
177:
464:
123:
for the category of modules. This term can be ambiguous since it could also refer to a category with a
89:
390:
173:
124:
54:
692:
265:
50:
642:
58:
598:
8:
289:
232:
70:
479:
438:
422:
311:
140:
35:
584:
572:
444:
366:
144:
103:
but that category is equivalent to the category of left (or right) modules over the
594:
449:
307:
169:
158:
152:
136:
31:
474:
454:
378:
162:
119:
638:
634:
499:
350:
681:
224:
426:
285:
394:
192:
296:
626:
612:
429:
also has enough injectives (though not always enough projectives).
66:
259:
82:
516:
trivially since any module is a quotient of a free module.
99:
One can also define the category of bimodules over a ring
441:(the important invariant of the category of modules.)
227:of the category of modules over a commutative ring
165:exist in the categories of left and right modules.
679:
250:-mod is exactly a finitely presented module).
662:
135:The categories of left and right modules are
253:
583:. Vol. 5 (second ed.). Springer.
405:which has as its objects the vector spaces
155:of the category of modules over some ring.
669:
655:
151:states every abelian category arises as a
577:Categories for the Working Mathematician
571:
562:
525:
302:as morphisms. Since vector spaces over
14:
680:
621:
553:
537:
187:
363:dimension theorem for vector spaces
306:(as a field) are the same thing as
24:
416:
25:
709:
606:
625:
613:http://ncatlab.org/nlab/show/Mod
460:Category of graded vector spaces
191:
176:⊗, the category of modules is a
111:(or over the opposite of that).
563:Dummit, David; Foote, Richard.
547:
554:Bourbaki. "Algèbre linéaire".
531:
519:
510:
492:
88:, it is the same thing as the
13:
1:
688:Categories in category theory
581:Graduate Texts in Mathematics
485:
130:
96:is defined in a similar way.
641:. You can help Knowledge by
353:concerns the description of
149:Mitchell's embedding theorem
77:-modules. For example, when
7:
470:Category of representations
432:
178:symmetric monoidal category
10:
714:
620:
465:Category of abelian groups
377:correspond exactly to the
257:
183:
117:Some authors use the term
90:category of abelian groups
500:"module category in nLab"
254:Category of vector spaces
246:(a compact object in the
174:tensor product of modules
94:category of right modules
413:is any cardinal number.
342:), the category of left
139:. These categories have
125:monoidal-category action
43:category of left modules
27:Category in mathematics
637:-related article is a
528:, Ch. 10, Theorem 38.
325:is a special case of
698:Linear algebra stubs
172:, together with the
71:module homomorphisms
367:isomorphism classes
361:. For example, the
233:associative algebra
575:(September 1998).
573:Mac Lane, Saunders
526:Dummit & Foote
480:Morita equivalence
439:Algebraic K-theory
423:sheaves of modules
333:(some authors use
275:(some authors use
203:. You can help by
141:enough projectives
137:abelian categories
105:enveloping algebra
650:
649:
445:Category of rings
221:
220:
159:Projective limits
145:enough injectives
16:(Redirected from
705:
671:
664:
657:
629:
622:
602:
568:
565:Abstract Algebra
559:
541:
535:
529:
523:
517:
514:
508:
507:
496:
450:Derived category
421:The category of
379:cardinal numbers
295:as objects, and
216:
213:
195:
188:
170:commutative ring
163:inductive limits
153:full subcategory
21:
713:
712:
708:
707:
706:
704:
703:
702:
678:
677:
676:
675:
618:
609:
591:
550:
545:
544:
536:
532:
524:
520:
515:
511:
498:
497:
493:
488:
475:Change of rings
455:Module spectrum
435:
419:
417:Generalizations
341:
283:
262:
256:
217:
211:
208:
201:needs expansion
186:
133:
120:module category
81:is the ring of
28:
23:
22:
18:Module category
15:
12:
11:
5:
711:
701:
700:
695:
693:Linear algebra
690:
674:
673:
666:
659:
651:
648:
647:
635:linear algebra
630:
616:
615:
608:
607:External links
605:
604:
603:
589:
569:
560:
549:
546:
543:
542:
530:
518:
509:
490:
489:
487:
484:
483:
482:
477:
472:
467:
462:
457:
452:
447:
442:
434:
431:
418:
415:
365:says that the
351:linear algebra
337:
279:
255:
252:
244:compact object
231:is exactly an
219:
218:
198:
196:
185:
182:
132:
129:
26:
9:
6:
4:
3:
2:
710:
699:
696:
694:
691:
689:
686:
685:
683:
672:
667:
665:
660:
658:
653:
652:
646:
644:
640:
636:
631:
628:
624:
623:
619:
614:
611:
610:
600:
596:
592:
590:0-387-98403-8
586:
582:
578:
574:
570:
566:
561:
557:
552:
551:
539:
534:
527:
522:
513:
505:
501:
495:
491:
481:
478:
476:
473:
471:
468:
466:
463:
461:
458:
456:
453:
451:
448:
446:
443:
440:
437:
436:
430:
428:
424:
414:
412:
408:
404:
400:
396:
392:
388:
384:
380:
376:
372:
368:
364:
360:
356:
352:
347:
345:
340:
336:
332:
328:
324:
320:
316:
313:
309:
305:
301:
299:
294:
291:
287:
286:vector spaces
282:
278:
274:
270:
267:
261:
251:
249:
245:
240:
238:
234:
230:
226:
225:monoid object
215:
206:
202:
199:This section
197:
194:
190:
189:
181:
179:
175:
171:
166:
164:
160:
156:
154:
150:
146:
142:
138:
128:
126:
122:
121:
116:
112:
110:
106:
102:
97:
95:
91:
87:
84:
80:
76:
73:between left
72:
68:
64:
60:
57:are all left
56:
52:
48:
44:
40:
37:
33:
19:
643:expanding it
632:
617:
576:
564:
555:
548:Bibliography
533:
521:
512:
503:
494:
427:ringed space
420:
410:
406:
402:
398:
386:
382:
374:
370:
358:
354:
348:
343:
338:
334:
330:
326:
322:
318:
314:
303:
300:-linear maps
297:
292:
280:
276:
272:
268:
263:
247:
241:
236:
228:
222:
209:
205:adding to it
200:
167:
157:
134:
118:
114:
113:
108:
100:
98:
93:
85:
78:
74:
62:
46:
42:
38:
29:
504:ncatlab.org
395:subcategory
381:, and that
682:Categories
599:0906.18001
486:References
391:equivalent
346:-modules.
284:) has all
258:See also:
242:See also:
212:March 2023
131:Properties
65:and whose
34:, given a
310:over the
67:morphisms
538:Bourbaki
433:See also
409:, where
349:Much of
266:category
83:integers
69:are all
51:category
556:Algèbre
425:over a
393:to the
308:modules
288:over a
260:FinVect
184:Objects
168:Over a
59:modules
55:objects
49:is the
32:algebra
597:
587:
540:, § 6.
92:. The
53:whose
41:, the
633:This
290:field
235:over
115:Note:
61:over
45:over
639:stub
585:ISBN
403:Vect
387:Vect
375:Vect
359:Vect
323:Vect
312:ring
277:Vect
273:Vect
264:The
161:and
143:and
36:ring
595:Zbl
397:of
389:is
369:in
335:Mod
331:Mod
207:.
107:of
30:In
684::
593:.
579:.
502:.
317:,
239:.
223:A
180:.
147:.
127:.
670:e
663:t
656:v
645:.
601:.
567:.
558:.
506:.
411:n
407:K
401:-
399:K
385:-
383:K
373:-
371:K
357:-
355:K
344:R
339:R
329:-
327:R
321:-
319:K
315:K
304:K
298:K
293:K
281:K
271:-
269:K
248:R
237:R
229:R
214:)
210:(
109:R
101:R
86:Z
79:R
75:R
63:R
47:R
39:R
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
