Invariant factor

The invariant factors of a module over a principal ideal domain occur in one form of the structure theorem for finitely generated modules over a principal ideal domain.

If $R$ is a PID and $M$ a finitely generated $R$-module, then

:$Mcong\; R^roplus\; R/(a\_1)oplus\; R/(a\_2)opluscdotsoplus\; R/(a\_m)$

for some $rinmathbb\{Z\}\_0^+$ and nonzero elements $a\_1,ldots,a\_min\; R$ for which $a\_1midcdotsmid\; a\_m$. The nonnegative integer $r$ is called the "free rank" or "Betti number" of the module $M$, while $a\_1,ldots,a\_m$ are the "invariant factors" of $M$ and are unique up to associatedness.

The invariant factors of a 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.

ee also

* Elementary divisors

References

* Chap.8, p.128.
* Chap.III.7, p.153.