Is there an example where the Artin Rees lemma fails if we do not assume that $R$ is an integral domain?

Question

I have the following statement:

Artin Rees Lemma Let $R$ be a Noetherian integral domain, $M\in Mod_R$ a finitely generated $R$-module, $N\subset M$ a sub-$R$-module and $I\subset R$ an ideal. Then there exists $c\in \Bbb{N}_{>0}$ such that for all $n\geq c$ $$I^nM\cap N=I^{n-c}\left(I^cM\cap N\right)$$

Now I wanted to think about a simple counterexample which shows that the assumption of $R$ being an integral domain is really needed. Unfortunately I couldn't come up with one.

It would be nice if you could give me some help.

Thanks!

Answer 1

There is no such example; the theorem is true for any Noetherian (commutative) ring, see here: https://en.wikipedia.org/wiki/Artin%E2%80%93Rees_lemma

p.s. $M$ should be finitely generated