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