Elementary divisors: Difference between revisions

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{{Short description|Algebraic formula}}

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]].

If <math>R</math> is a [[Principal ideal ___domain|PID]] and <math>M</math> a [[finitely generated module|finitely generated]] <math>R</math>-module, then ''M'' is [[isomorphic]] to a unique finite [[direct sum of modules|direct sum]] of the form

::<math>M\cong R^r\oplus \bigoplus_{i=1}^l R/(q_i) \qquad\text{with }r,l\geq0</math>,

:where the <math>(q_i)</math> are [[primary ideal]]s (in particular <math>(q_i)\neq R</math>) such that <math>(q_1)\supset (q_2)\supset \cdots</math>.

where the <math>(q_i)</math> are nonzero [[primary ideal]]s.

The ideals <math>(q_i)</math> are unique; the elements <math>q_i</math> are unique up to [[associatedness]], and are called the ''elementary divisors''. Note that in a PID, primary ideals are powers of primes, so the elementary divisors <math>(q_i)=(p_i^{r_i}) = (p_i)^{r_i}</math>. The nonnegative integer <math>r</math> is called the ''free rank'' or ''Betti number'' of the module <math>M</math>.

The list of primary ideals <math>is unique [[up to]] order (q_i)</math>but area uniquegiven 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, the nonzero primary ideals are powers of primesprime ideals, so the elementary divisors can be written as powers <math>(q_i)=(p_i^{r_i}) = (p_i)^{r_i}</math>. of [[irreducible element]]s. The nonnegative [[integer]] <math>r</math> is called the ''free rank'' or ''Betti number'' of the module <math>M</math>.

The elementary divisors of a [[Matrix (mathematics)|matrix]] over a PID occur in the [[Smith normal form]] and provide a means of computing the structure of a module from a set of generators and relations.

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''^''k''}} 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 (ring theory)|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''. [[Converse (logic)|Converse]]ly, 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''^''k''}} 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 [[empty set|non-empty]], repeat to find the invariant factors before it.

==See also==

* [[Invariant factors]]

* [[Smith normal form]]

==References==

* {{cite book | author=B. Hartley | authorlink=Brian Hartley | coauthorsauthor2=T.O. Hawkes | title=Rings, modules and linear algebra | publisher=Chapman and Hall | year=1970 | isbn=0-412-09810-5 }} Chap.11, p.182.

* Chap. III.7, p.153 of {{Lang Algebra|edition=3}}

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