Invariant factor

One form of the structure theorem for finitely generated modules over a principal ideal domain states that 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 $r\in \mathbb {Z} _{0}^{+}$ and nonzero elements $a_{1},\ldots ,a_{m}\in R$ for which $a_{1}\mid \cdots \mid a_{m}$. 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.