Invariant factor

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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 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 integer r\geq0 and a (possibly empty) list of nonzero elements a_1,\ldots,a_m\in R for which a_1 \mid a_2 \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.

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.

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