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.

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 Pub. Co., ISBN 978-0-201-55540-0, Zbl 0848.13001

This article is issued from Wikipedia - version of the 5/12/2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Invariant factor

See also

References