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 unique sum of the form

${\displaystyle M\cong R^{r}\oplus \bigoplus _{i}R/(q_{i})}$

where the ${\displaystyle (q_{i})}$ are primary ideals (in particular ${\displaystyle (q_{i})\neq R}$) such that ${\displaystyle (q_{1})\supset (q_{2})\supset \cdots }$.

The ideals ${\displaystyle (q_{i})}$ are unique; the elements ${\displaystyle q_{i}}$ 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 ${\displaystyle (q_{i})=(p_{i}^{r_{i}})=(p_{i})^{r_{i}}}$. The nonnegative integer ${\displaystyle r}$ is called the free rank or Betti number of the module ${\displaystyle M}$.

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.

See also

• Invariant factors

References

• B. Hartley (1970). Rings, modules and linear algebra. Chapman and Hall. ISBN 0-412-09810-5. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) Chap.11, p.182.
• Chap. III.7, p.153 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001