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 completionharv|Atiyah|MacDonald|1969|pp=107–109.

tatement 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

*Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969

External links

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