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
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.
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