 Artin–Rees lemma

If $R$ is Noetherian, $I$ is an ideal, and $N\subseteq M$ are finitely generated modules, then some $k$ satisfiesfor every $n\ge k$.

The Artin–Rees lemma is a fundamental result in commutative algebra, particularly in the theory of Noetherian rings and ideals. It provides a way to control the behavior of ideals under powers and the localization of modules over a Noetherian ring.