Invariant factor

The fundamental theorem of finitely generated modules over principal ideal domains states that if $R$ is a PID and $M$ a finitely generated $R$-module, then $$M\cong R^r\oplus R/(a_1)\oplus R/(a_2)\oplus\cdots\odots R/(a_m)$$ 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 multiplication by a unit in $R$.

This result is applied when considering transformations from a vector space $V$ over a field $F$ to itself. Objects such as the characteristic polynomial then lie in the ring $F[x]$ of polynomials over the field $F$, which is a PID. We thus consider $V$ as an $F[x]$-module to obtain a space $V$ is isomorphic to.

The fundamental theorem of finitely generated abelian groups follows from the above as result as a corollary. The term invariant factor is used analogously in the decomposition of such finitely generated abelian groups into products of copies of $\mathbb{Z}$ and copies of cyclic groups.