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. See also
 Elementary divisors

References
   Chap.8, p.128. Chapter III.7, p.153 of 

Module theory