invariant factor

The invariant factors of a 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 $R$ is a PID and $M$ a finitely generated $R$-module, then

:$M\backslash cong\; R^r\backslash oplus\; R/(a\_1)\backslash oplus\; R/(a\_2)\backslash oplus\backslash cdots\backslash oplus\; R/(a\_m)$

for some integer $r\backslash geq0$ and a (possibly empty) list of nonzero elements $a\_1,\backslash ldots,a\_m\backslash in\; R$ for which $a\_1\; \backslash mid\; a\_2\; \backslash mid\; \backslash cdots\; \backslash mid\; a\_m$. The nonnegative integer $r$ is called the free rank or Betti number of the module $M$, while $a\_1,\backslash ldots,a\_m$ 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.