# 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 is a PID and a finitely generated -module, then

for some integer and a (possibly empty) list of nonzero elements for which . The nonnegative integer is called the *free rank* or *Betti number* of the module , while are the *invariant factors* of 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

## 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

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