Elementary divisors: Difference between revisions

Revision as of 04:05, 17 July 2013

Revision as of 14:30, 7 December 2013

Line 3:

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 sum of the form

::<math>M\cong R^r\oplus \bigoplus_i R/(q_i)</math>

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

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 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.