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17:
446:
652:
172:
683:
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is generated as a principal ideal by 1, but it is also generated by, say, a minimal generating set
542:
A more refined information is obtained if one considers the relations between the generators; see
565:
376:
344:-th component of the direct sum. (Coincidentally, since a generating set always exists, e.g.
586:"ac.commutative algebra – Existence of a minimal generating set of a module – MathOverflow"
528:
39:
8:
436:
372:
100:
46:
625:
349:
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of all submodules containing the set). The set Γ is then said to generate
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524:
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353:
31:
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248:
89:
321:{\displaystyle \bigoplus _{g\in \Gamma }R\to M,\,r_{g}\mapsto r_{g}g,}
107:
is an ideal that has a generating set consisting of a single element.
61:
617:
390:
of a minimal generating set need not be an invariant of the module;
585:
403:
103:, which are the submodules of the ring itself. In particular, a
53:
122:-linear combination of some elements of Γ; i.e., for each
72:
itself (the smallest submodule containing a subset is the
375:, then a minimal generating set is the same thing as a
512:{\displaystyle \dim _{k}M/mM=\dim _{k}M\otimes _{R}k}
449:
259:
175:
511:
443:has a minimal generating set whose cardinality is
320:
236:
110:Explicitly, if Γ is a generating set of a module
84:is generated by the identity element 1 as a left
670:
406:of the numbers of the generators of the module.
237:{\displaystyle x=r_{1}g_{1}+\cdots +r_{m}g_{m}.}
653:
383:, there may exist no minimal generating set.
92:generating set, then a module is said to be
359:A generating set of a module is said to be
660:
646:
288:
402:uniquely determined by a module is the
14:
671:
527:, then this minimal generating set is
348:itself, this shows that a module is a
612:
367:of the set generates the module. If
88:-module over itself. If there is a
24:
271:
25:
695:
616:
435:finitely generated module. Then
601:Dummit, David; Foote, Richard.
247:Put in another way, there is a
578:
299:
279:
13:
1:
571:
544:Free presentation of a module
632:. You can help Knowledge by
7:
549:
10:
700:
611:
556:Countably generated module
18:Generating set of an ideal
114:, then every element of
80:. For example, the ring
379:. Unless the module is
60:such that the smallest
684:Abstract algebra stubs
628:-related article is a
566:Invariant basis number
513:
340:for an element in the
322:
238:
27:Concept in mathematics
514:
323:
239:
535:is free). See also:
529:linearly independent
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257:
173:
537:Minimal resolution
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381:finitely generated
356:, a useful fact.)
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275:
234:
94:finitely generated
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16:(Redirected from
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679:Abstract algebra
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626:abstract algebra
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603:Abstract Algebra
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590:mathoverflow.net
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437:Nakayama's lemma
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99:This applies to
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331:where we wrote
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118:is a (finite)
36:generating set
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365:proper subset
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634:expanding it
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130:, there are
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85:
81:
77:
74:intersection
69:
65:
57:
49:
42:
35:
29:
561:Flat module
388:cardinality
354:free module
32:mathematics
673:Categories
572:References
439:says that
415:local ring
249:surjection
498:⊗
491:
461:
300:↦
280:→
272:Γ
269:∈
262:⨁
206:⋯
62:submodule
550:See also
398:}. What
350:quotient
404:infimum
361:minimal
157:, ...,
137:, ...,
45:over a
38:Γ of a
363:if no
101:ideals
90:finite
54:subset
40:module
624:This
519:. If
417:with
413:be a
396:{2, 3
377:basis
373:field
371:is a
352:of a
52:is a
630:stub
531:(so
525:flat
431:and
424:and
409:Let
386:The
150:and
47:ring
34:, a
523:is
482:dim
452:dim
146:in
126:in
64:of
56:of
30:In
675::
588:.
546:.
539:.
400:is
96:.
661:e
654:t
647:v
636:.
605:.
592:.
533:M
521:M
507:k
502:R
494:M
486:k
478:=
475:M
472:m
468:/
464:M
456:k
441:M
433:M
429:k
422:m
411:R
392:Z
369:R
346:M
342:g
337:g
333:r
316:,
313:g
308:g
304:r
295:g
291:r
286:,
283:M
277:R
266:g
232:.
227:m
223:g
217:m
213:r
209:+
203:+
198:1
194:g
188:1
184:r
180:=
177:x
163:m
159:g
155:1
152:g
148:R
143:m
139:r
135:1
132:r
128:M
124:x
120:R
116:M
112:M
86:R
82:R
78:M
70:M
66:M
58:M
50:R
43:M
20:)
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