Lemma 10.51.5. Let $R$ be a Noetherian ring. Let $I \subset R$ be an ideal. Let $M$ be a finite $R$-module. Let $N = \bigcap _ n I^ n M$.

For every prime $\mathfrak p$, $I \subset \mathfrak p$ there exists a $f \in R$, $f \not\in \mathfrak p$ such that $N_ f = 0$.

If $I$ is contained in the Jacobson radical of $R$, then $N = 0$.

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