 Krull intersection theorem

Let $R$ be a Noetherian local ring, $I$ an ideal contained in its maximal ideal, and $M$ a finitely generated module. The Krull intersection theorem statesIn particular ${\bigcap }_n{I}^n=0$ by taking $M=R$. It says that the I-adic topology is separated on every finite module.

For a proof, let $N={\bigcap }_n{I}^{n}M$. It is finitely generated because $M$ is Noetherian. The Artin-Rees lemma gives, for some $k$, $I^{k+1}M\cap N=I(I^{k}M\cap N)$. Both intersections equal $N$, so $N=IN$. The Nakayama lemma then gives $N=0$. Finite generation is essential to this argument; it cannot be omitted merely because the scalar ring is Noetherian.