Invariant factor: Difference between revisions

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The '''invariant factors''' 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

:<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.

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==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.8, p.128.

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

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