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 ${\displaystyle R}$ is a PID and ${\displaystyle M}$ a finitely generated ${\displaystyle R}$-module, then

${\displaystyle M\cong R^{r}\oplus R/(a_{1})\oplus R/(a_{2})\oplus \cdots \oplus R/(a_{m})}$

for some integer ${\displaystyle r\geq 0}$ and a (possibly empty) list of nonzero elements ${\displaystyle a_{1},\ldots ,a_{m}\in R}$ for which ${\displaystyle a_{1}\mid a_{2}\mid \cdots \mid a_{m}}$. The nonnegative integer ${\displaystyle r}$ is called the free rank or Betti number of the module ${\displaystyle M}$, while ${\displaystyle a_{1},\ldots ,a_{m}}$ are the invariant factors of ${\displaystyle M}$ 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.