Artin-Rees lemma

This article is about the statement of a simple but indispensable lemma in commutative algebra
View other indispensable lemmata

Statement

Suppose $A$ is a Noetherian commutative unital ring and $M$ is a finitely generated $A$ -module and $N$ a submodule of $M$ .

Suppose:

$M = M_{0} \supset M_{1} \supset M_{2} \supset \dots$

is an essentially $I$ -adic filtration (in other words, there exists $n_{0}$ such that for all $n \geq n_{0}$ , $I M_{n} = M_{n + 1}$ ).

Then the filtration of $N$ given by:

$N = N \cap M_{0} \supset N \cap M_{1} \supset N \cap M_{2} \supset \dots$

is also essentially $I$ -adic.