Invariant factor: Difference between revisions

Revision as of 20:10, 12 September 2013

Revision as of 06:31, 15 December 2013

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:<math>M\cong R^r\oplus R/(a_1)\oplus R/(a_2)\oplus\cdots\oplus R/(a_m)</math>

for some integer <math>r\in\mathbb{Z}_0^+geq0</math> and a (possibly empty) list of nonzero elements <math>a_1,\ldots,a_m\in R</math> for which <math>a_1 \mid a_2 \mid \cdots \mid a_m</math>. The nonnegative integer <math>r</math> is called the ''free rank'' or ''Betti number'' of the module <math>M</math>, while <math>a_1,\ldots,a_m</math> are the ''invariant factors'' of <math>M</math> and are unique up to [[associatedness]].

The invariant factors 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.