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 Failed to parse (unknown function "\odots"): {\displaystyle 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.