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


This article is issued from Wikipedia - version of the 5/12/2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.