Invariant factor

The invariant factors 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

$M\cong R^{r}\oplus R/(a_{1})\oplus R/(a_{2})\oplus \cdots \oplus R/(a_{m})$

for some $r\in \mathbb {Z} _{0}^{+}$ and nonzero elements $a_{1},\ldots ,a_{m}\in R$ for which $a_{1}\mid \cdots \mid a_{m}$. The nonnegative integer $r$ is called the free rank or Betti number of the module $M$, while $a_{1},\ldots ,a_{m}$ are the invariant factors of $M$ and are unique up to associatedness.

The invariant factors 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

• Elementary divisors

References

• B. Hartley and T.O. Hawkes, Rings, modules and linear algebra, Chapman and Hall, 1970, ISBN 0-412-09810-5. Chap.8, p.128.
• Serge Lang, Algebra (3 ed), Addison-Wesley, 1993, ISBN 0-201-55540-9. Chap.III.7, p.153.