573:
433:
of the module by decomposing each of them as far as possible into pairwise relatively prime (non-unit) factors, which will be powers of irreducible elements. This decomposition corresponds to maximally decomposing each submodule corresponding to an invariant factor by using the
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36:
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of elementary divisors, the invariant factors can be found, starting from the final one (which is a multiple of all others), as follows. For each irreducible element
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60:
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302:. Note that in a PID, the nonzero primary ideals are powers of prime ideals, so the elementary divisors can be written as powers
263:
The list of primary ideals is unique up to order (but a given ideal may be present more than once, so the list represents a
638:
600:
435:
633:
63:
32:
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28:
429:
occurs among the elementary divisors. The elementary divisors can be obtained from the list of
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430:
210:{\displaystyle M\cong R^{r}\oplus \bigoplus _{i=1}^{l}R/(q_{i})\qquad {\text{with }}r,l\geq 0}
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structure theorem for finitely generated modules over a principal ideal domain
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The module is determined up to isomorphism by specifying its free rank
20:
472:, and multiply these powers together for all (classes of associated)
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is non-empty, repeat to find the invariant factors before it.
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411:, and for class of associated irreducible elements
542:(Third ed.), Reading, Mass.: Addison-Wesley,
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354:of irreducible elements. The nonnegative integer
466:, take the highest such power, removing it from
478:to give the final invariant factor; as long as
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110:is isomorphic to a finite sum of the form
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442:. Conversely, knowing the multiset
347:{\displaystyle q_{i}=p_{i}^{r_{i}}}
587:. You can help Knowledge (XXG) by
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516:Rings, modules and linear algebra
267:of primary ideals); the elements
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186:
35:(PID) occur in one form of the
243:
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183:
170:
1:
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566:
417:and each positive integer
436:Chinese remainder theorem
423:the number of times that
249:{\displaystyle (q_{i})}
583:-related article is a
533:Chap. III.7, p.153 of
514:; T.O. Hawkes (1970).
396:
368:
348:
294:are unique only up to
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250:
211:
161:
100:
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33:principal ideal domain
454:such that some power
397:
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298:, and are called the
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287:{\displaystyle q_{i}}
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86:a finitely generated
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639:Linear algebra stubs
518:. Chapman and Hall.
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300:elementary divisors
25:elementary divisors
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549:978-0-201-55540-0
499:Smith normal form
494:Invariant factors
431:invariant factors
395:{\displaystyle M}
367:{\displaystyle r}
190:
99:{\displaystyle R}
79:{\displaystyle M}
55:{\displaystyle R}
16:Algebraic formula
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530:Chap.11, p.182.
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5:
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581:linear algebra
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531:
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506:
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382:of the module
374:is called the
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296:associatedness
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258:primary ideals
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106:-module, then
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15:
13:
10:
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3:
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635:
634:Module theory
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525:0-412-09810-5
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297:
279:
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266:
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115:
114:
113:
112:
111:
109:
93:
73:
65:
49:
40:
38:
34:
30:
26:
22:
589:expanding it
578:
563:
539:
515:
480:
474:
468:
462:
456:
450:
444:
439:
425:
419:
413:
407:
404:
380:Betti number
379:
375:
299:
262:
256:are nonzero
222:
107:
41:
24:
18:
536:Lang, Serge
628:Categories
558:0848.13001
512:B. Hartley
505:References
460:occurs in
223:where the
189:with
376:free rank
202:≥
143:⨁
139:⊕
126:≅
538:(1993),
488:See also
265:multiset
540:Algebra
31:over a
21:algebra
556:
546:
522:
29:module
23:, the
579:This
62:is a
27:of a
585:stub
544:ISBN
520:ISBN
438:for
66:and
554:Zbl
378:or
64:PID
42:If
19:In
630::
552:,
402:.
260:.
39:.
616:e
609:t
602:v
591:.
528:.
481:M
475:p
469:M
463:M
457:p
451:p
445:M
440:R
426:p
420:k
414:p
408:r
390:M
362:r
338:i
334:r
328:i
324:p
320:=
315:i
311:q
280:i
276:q
244:)
239:i
235:q
231:(
217:,
205:0
199:l
196:,
193:r
184:)
179:i
175:q
171:(
167:/
163:R
158:l
153:1
150:=
147:i
134:r
130:R
123:M
108:M
94:R
74:M
50:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
