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


 * $$M\cong R^r\oplus R/(a_1)\oplus R/(a_2)\oplus\cdots\oplus R/(a_m)$$

for some integer $$r\geq0$$ and a (possibly empty) list of nonzero elements $$a_1,\ldots,a_m\in R$$ for which $$a_1 \mid a_2 \mid \cdots \mid a_m$$. 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.