Lemma 10.50.4 (Krull's intersection theorem). Let $R$ be a Noetherian local ring. Let $I \subset R$ be a proper ideal. Let $M$ be a finite $R$-module. Then $\bigcap _{n \geq 0} I^ nM = 0$.

**Proof.**
Let $N = \bigcap _{n \geq 0} I^ nM$. Then $N = I^ nM \cap N$ for all $n \geq 0$. By the Artin-Rees Lemma 10.50.2 we see that $N = I^ nM \cap N \subset IN$ for some suitably large $n$. By Nakayama's Lemma 10.19.1 we see that $N = 0$.
$\square$

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