Invariant factor

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

:Mcong R^roplus R/(a_1)oplus R/(a_2)opluscdotsoplus R/(a_m)

for some rinmathbb{Z}_0^+ and nonzero elements a_1,ldots,a_min R for which a_1midcdotsmid 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.

ee also

* Elementary divisors

References

* Chap.8, p.128.
* Chap.III.7, p.153.


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