Solution
= Solution
Let $N=\bigcap_{n\geq0}I^nM$. Apply Artin--Rees to $N\subseteq M$. Since $N\subseteq I^nM$ for every $n$, stability gives
$$
N=I^{n-\ell}N
$$
for all sufficiently large $n$, and in particular $N=IN$. The module $N$ is finitely generated because $M$ is Noetherian. The determinant trick applied to a finite generating set of $N=IN$ produces $a\in I$ with
$$
(1-a)N=0.
$$
Taking $x=-a\in I$ yields
$$
\boxed{(1+x)N=0,}
$$
which is <Krull intersection theorem>.