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 R is a PID and M a finitely generated R -module, then

for some integer r ≥ 0 and a (possibly empty) list of nonzero elements a 1 , … , a m ∈ R for which a 1 ∣ a 2 ∣ ⋯ ∣ a m . The nonnegative integer r is called the free rank or Betti number of the module M , while a 1 , … , a m are the invariant factors of M 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.