= Krull intersection theorem
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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> states
$$
\bigcap_{n\geq0}I^nM=0.
$$
In 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 $N=\bigcap_n I^nM$. It is finitely generated because $M$ is Noetherian. The <Artin-Rees lemma> gives, for some $k$, $I^{k+1}M\cap N=I(I^kM\cap N)$. 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.
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