Artin–Rees lemma

In mathematics, the Artin–Rees lemma (also known as the Artin–Rees theorem) is a result in the theory of rings and modules. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work. The result is used to prove the exactness property of completion(Atiyah & MacDonald 1969, pp. 107–109).

Statement of the result

Let I be an ideal in a Noetherian ring R; let M be a finitely generated R-module and let N a submodule of M. Then there exists an integer k ≥ 1 so that, for n ≥ k,

$I^{n} M \cap N = I^{n - k} ((I^{k} M) \cap N).$

References

• Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8 

External links

• Artin-Rees Theorem on PlanetMath