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| In [[algebra]], the '''elementary divisors''' of a [[module (mathematics)|module]] over a [[principal ideal domain]] (PID) occur in one form of the [[structure theorem for finitely generated modules over a principal ideal domain]].
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| If <math>R</math> is a [[Principal ideal domain|PID]] and <math>M</math> a finitely generated <math>R</math>-module, then ''M'' is isomorphic to a finite sum of the form
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| ::<math>M\cong R^r\oplus \bigoplus_{i=1}^l R/(q_i) \qquad\text{with }r,l\geq0</math>
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| :where the <math>(q_i)</math> are [[primary ideal]]s (in particular <math>(q_i)\neq R</math>). | |
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| The list of primary ideals is unique up to order (but a same ideal may be present more than once, so the list represents a [[multiset]] of primary ideals); the elements <math>q_i</math> are unique only up to [[associatedness]], and are called the ''elementary divisors''. Note that in a PID, primary ideals are powers of prime ideals, so the elementary divisors can be written as powers <math>q_i=p_i^{r_i}</math> of irreducible elements. The nonnegative integer <math>r</math> is called the ''free rank'' or ''Betti number'' of the module <math>M</math>.
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| The module is determined up to isomorphism by specifying its free rank {{math|''r''}}, and for class of associated irreducible elements {{math|''p''}} and each positive integer {{math|''k''}} the number of times that {{math|''p''<sup>''k''</sup>}} occurs among the elementary divisors. The elementary divisors can be obtained from the list of [[invariant factors]] 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 [[Chinese remainder theorem#Statement_for_principal_ideal_domains|Chinese remainder theorem]] for ''R''. Conversely, knowing the multiset {{math|''M''}} 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 {{math|''p''}} such that some power {{math|''p''<sup>''k''</sup>}} occurs in {{math|''M''}}, take the highest such power, removing it from {{math|''M''}}, and multiply these powers together for all (classes of associated) {{math|''p''}} to give the final invariant factor; as long as {{math|''M''}} is non-empty, repeat to find the invariant factors before it. | |
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| ==See also== | |
| * [[Invariant factors]]
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| ==References==
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| * {{cite book | author=B. Hartley | authorlink=Brian Hartley | coauthors=T.O. Hawkes | title=Rings, modules and linear algebra | publisher=Chapman and Hall | year=1970 | isbn=0-412-09810-5 }} Chap.11, p.182.
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| * Chap. III.7, p.153 of {{Lang Algebra|edition=3}}
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| [[Category:Module theory]]
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| {{Abstract-algebra-stub}}
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