Artin–Rees lemma

In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. 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.

One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion (Atiyah & MacDonald 1969, pp. 107–109).

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,

The lemma immediately follows from the fact that R is Noetherian once necessary notions and notations are set up.

For any ring R and an ideal I in R, we set ${\displaystyle B_{I}R=\textstyle \bigoplus _{n=0}^{\infty }I^{n}}$ (B for blow-up.) We say a decreasing sequence of submodules ${\displaystyle M=M_{0}\supset M_{1}\supset M_{2}\supset \cdots }$ is an I-filtration if ${\displaystyle IM_{n}\subset M_{n+1}}$; moreover, it is stable if ${\displaystyle IM_{n}=M_{n+1}}$ for sufficiently large n. If M is given an I-filtration, we set ${\displaystyle B_{I}M=\textstyle \bigoplus _{n=0}^{\infty }M_{n}}$; it is a graded module over ${\displaystyle B_{I}R}$.

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