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:

$M=M_0\supset M_1\supset M_2\supset \dots$

is an essentially $I$-adic filtration (in other words, there exists $n_0$ such that for all $n\ge n_0$, $I{M}_n=M_{n+1}$).

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.