This well known Lemma about $I$-stable filtrations asserts:
Lemma (Artin-Rees) Let $A$ be a Noetherian ring and $E$ a finitely generated $A$-module. Let $F$ be a submodule of $E$ and $\{E_i\}$ an $I$-stable filtration. Then the induced filtration on $F$ is $I$-stable.
One of its immediate corollaries is the following:
Corollary Let $A$ be a Noetherian ring and $I$ an ideal. Then the following equality holds: $$I \left( \bigcap_{i=0}^{\infty} I^i\right)=\left( \bigcap_{i=0}^{\infty} I^i\right)$$
I'm looking for a counterexample to the Artin-Rees Lemma in one of the two forms above, when the ring A is not Noetherian.
Does anyone have some ideas or references?
Thank you!