# 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 Template:Harv.

## Statement

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,

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

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

Now, let M be a R-module with the I-filtration $M_{i}$ by finitely generated R-modules. We make an observation

$B_{I}M$ is a finitely generated module over $B_{I}R$ if and only if the filtration is I-stable.
$f=\sum a_{ij}g_{ij},\quad a_{ij}\in I^{n-j}$ ## Proof of Krull's intersection theorem

Besides the use in completion of a ring, a typical application of the lemma is the proof of the Krull's intersection theorem, which says: $\cap _{1}^{\infty }I^{n}=0$ for a proper ideal I in a Noetherian local ring. By the lemma applied to the intersection N, we find k such that for $n\geq k$ ,

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