Artin-Rees lemma

This article is about the statement of a simple but indispensable lemma in commutative algebra
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This article defines a result where the base ring (or one or more of the rings involved) is Noetherian
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Statement

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

Suppose:

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

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

Then the filtration of $N$ given by:

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

is also essentially $I$-adic.