Deep enough in an $I$-adic filtration, intersection with a submodule is a fixed shifted $I$-adic filtration.Suggest a slogan
Let \(R\) be a commutative Noetherian ring, let \(I\subseteq R\) be an ideal, let \(M\) be a finitely generated \(R\)-module, and let \(N\subseteq M\) be a submodule. Then there is an integer \(c\ge0\) such that for every \(n\ge c\), \[ I^nM\cap N=I^{n-c}\qty{I^cM\cap N}. \]
What is the Artin-Rees lemma?