Recall the following theorem from commutative algebra:
Theorem (Artin-Rees Lemma). Suppose $M$ and $N$ are finitely generated modules over a Noetherian ring $A$, and $N$ is a submodule of $M$. Let $I$ be an ideal of $A$. For every stable $I$-filtration $\{M_n\}_n$ on $M$, $$ I(M_n\cap N)=M_{n+1}\cap N $$ for sufficiently large $n$. Hence, $\{M_n\cap N\}_n$ is a stable $I$-filtration on $N$.
In the special case where $M=A$, $N=\bigcap_{n\ge0}I^n$, and $M_n=I^n$, the Artin-Rees lemma asserts that $$ I\bigcap_{n\ge0}I^n=\bigcap_{n\ge0}I^n \, . \tag{$\star$}\label{*} $$ It seems that this equality does not hold in general for non-Noetherian rings $A$; however, I cannot think of a counter-example. I tried letting $A$ be the ring of smooth functions on $\mathbb R$, and $I$ be the ideal of functions vanishing at the origin, but I'm not sure that this works.
Question: is there a counter-example to \eqref{*} for non-Noetherian rings $A$?
