Elementary divisors: Difference between revisions

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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]].

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 unique finite 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>.

The list of primary ideals <math>is unique up to order (q_i)</math>but area uniquesame 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 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 elements. 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.

==See also==