Invariant factor

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 formula_1 is a PID and formula_2 a finitely generated formula_1-module, then
for some integer formula_5 and a (possibly empty) list of nonzero elements formula_6 for which formula_7. The nonnegative integer formula_8 is called the "free rank" or "Betti number" of the module formula_2, while formula_10 are the "invariant factors" of formula_2 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.