Invariant factor

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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.