# Invariant factor

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The **invariant factors** of a [module](/source/Module_(mathematics)) over a [principal ideal domain](/source/Principal_ideal_domain) (PID) occur in one form of the [structure theorem for finitely generated modules over a principal ideal domain](/source/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain).

If R {\displaystyle R} is a [PID](/source/Principal_ideal_domain) and M {\displaystyle M} a [finitely generated](/source/Finitely-generated_module) R {\displaystyle R} -module, then

- M ≅ R r ⊕ R / ( a 1 ) ⊕ R / ( a 2 ) ⊕ ⋯ ⊕ R / ( a m ) {\displaystyle M\cong R^{r}\oplus R/(a_{1})\oplus R/(a_{2})\oplus \cdots \oplus R/(a_{m})}

for some integer r ≥ 0 {\displaystyle r\geq 0} and a (possibly empty) list of nonzero elements a 1 , … , a m ∈ R {\displaystyle a_{1},\ldots ,a_{m}\in R} for which a 1 ∣ a 2 ∣ ⋯ ∣ a m {\displaystyle a_{1}\mid a_{2}\mid \cdots \mid a_{m}} . The nonnegative integer r {\displaystyle r} is called the *free rank* or *Betti number* of the module M {\displaystyle M} , while a 1 , … , a m {\displaystyle a_{1},\ldots ,a_{m}} are the *invariant factors* of M {\displaystyle M} and are unique up to [associatedness](/source/Associatedness).

The invariant factors of a [matrix](/source/Matrix_(mathematics)) over a PID occur in the [Smith normal form](/source/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](/source/Elementary_divisors)

## References

- [B. Hartley](/source/Brian_Hartley); T.O. Hawkes (1970). *Rings, modules and linear algebra*. Chapman and Hall. [ISBN](/source/ISBN_(identifier)) [0-412-09810-5](https://en.wikipedia.org/wiki/Special:BookSources/0-412-09810-5). Chap.8, p.128.

- Chapter III.7, p.153 of [Lang, Serge](/source/Serge_Lang) (1993), *[Algebra](/source/Algebra_(Lang))* (Third ed.), Reading, Mass.: Addison-Wesley, [ISBN](/source/ISBN_(identifier)) [978-0-201-55540-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-201-55540-0), [Zbl](/source/Zbl_(identifier)) [0848.13001](https://zbmath.org/?format=complete&q=an:0848.13001)

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