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

${\displaystyle M\cong R^{r}\oplus \bigoplus _{i=1}^{l}R/(q_{i})\qquad {\text{with }}r,l\geq 0}$

where the ${\displaystyle (q_{i})}$ are primary ideals (in particular ${\displaystyle (q_{i})\neq R}$).

The list of primary ideals is unique up to order (but a same 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, 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 ${\displaystyle M}$.

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