Elementary divisors

In algebra, the elementary divisors of a module over a principal ideal domain 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 "M" is isomorphic to a unique sum of the form:::where $q\_i\; eq\; 1$ and the $(q\_i)$ are primary ideals.

The ideals $(q\_i)$ are unique (up to order); the elements $q\_i$ 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 $(q\_i)=(p\_i^\{r\_i\})\; =\; (p\_i)^\{r\_i\}$. The nonnegative integer $r$ is called the "free rank" or "Betti number" of the module $M$.

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.

ee also

* Invariant factors

References

* Chap.11, p.182.
* Chap.III.7, p.153.