Invariant factor

The invariant factors 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 $R$ 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 integer $r\geq 0$ and a (possibly empty) list of nonzero elements $a_{1},\ldots ,a_{m}\in R$ for which $a_{1}\mid a_{2}\mid \cdots \mid a_{m}$. 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; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. ISBN 0-412-09810-5. Chap.8, p.128.
• Chapter III.7, p.153 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001