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.
a finitely generated
and a (possibly empty) list of nonzero elements
The nonnegative integer
is called the free rank or Betti number of the module
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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