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Invariant factor

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated June 6, 2026

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.

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This article is adapted from the Wikipedia article “Invariant factor”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Elementary divisors

In algebra, the elementary divisors 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 {\displaystyle R} is a PID and M {\displaystyle M} a finitely generated R {\displaystyle R} -module, then M is isomorphic to a finite direct sum of the form M ≅ R r ⊕ ⨁ i = 1 l R / ( q i ) with r , l ≥ 0 {\displaystyle M\cong R^{r}\oplus \bigoplus _{i=1}^{l}R/(q_{i})\qquad {\text{with }}r,l\geq 0} , where the ( q i ) {\displaystyle (q_{i})} are nonzero primary ideals.