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 ${\displaystyle R}$ is a PID and ${\displaystyle M}$ a finitely generated ${\displaystyle R}$-module, then M is isomorphic to a finite sum of the form

The list of primary ideals is unique up to order (but a given ideal may be present more than once, so the list represents a multiset of primary ideals); the elements ${\displaystyle q_{i}}$ are unique only up to associatedness, and are called the elementary divisors. Note that in a PID, the nonzero primary ideals are powers of prime ideals, so the elementary divisors can be written as powers ${\displaystyle q_{i}=p_{i}^{r_{i}}}$ of irreducible elements. The nonnegative integer ${\displaystyle r}$ is called the free rank or Betti number of the module ${\displaystyle M}$.

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