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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
215:
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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84:
60:
614:
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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
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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
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429:
occurs among the elementary divisors. The elementary divisors can be obtained from the list of
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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
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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542:(Third ed.), Reading, Mass.: Addison-Wesley,
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354:of irreducible elements. The nonnegative integer
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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}}}
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516:Rings, modules and linear algebra
267:of primary ideals); the elements
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35:(PID) occur in one form of the
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1:
504:
587:. You can help Knowledge by
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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).
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294:are unique only up to
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33:principal ideal domain
454:such that some power
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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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25:elementary divisors
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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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581:linear algebra
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382:of the module
374:is called the
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296:associatedness
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106:-module, then
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634:Module theory
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93:
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40:
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26:
22:
589:expanding it
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444:
439:
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419:
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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
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390:M
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320:=
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163:R
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150:=
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50:R
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