Elementary Divisors

In algebra, the elementary divisors 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 M is isomorphic to a unique sum of the form

where and the are primary ideals.

The ideals are unique (up to order); the elements are unique up to associatedness, and are called the elementary divisors. Note that in a PID, primary ideals are powers of primes, so the elementary divisors . The nonnegative integer is called the free rank or Betti number of the module .

The elementary divisors 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.