Revision as of 04:04, 17 July 2013 edit 61.95.189.159 (talk) The ideals q_1, q_2, etc. are not unique without the conditions that they form a chain under containment: for example Z/2 x Z/3 is Z/6 ← Previous edit		Revision as of 04:05, 17 July 2013 edit undo 61.95.189.159 (talk) Once the ideals are ordered by inclusion, they are unique Next edit →
Line 5: :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 ideals <math>(q_i)</math> are unique ~~(up to order)~~; 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.

Elementary divisors: Difference between revisions