Remark 10.51.6. Lemma 10.51.4 in particular implies that $\bigcap _ n I^ n = (0)$ when $I \subset R$ is a non-unit ideal in a Noetherian local ring $R$. More generally, let $R$ be a Noetherian ring and $I \subset R$ an ideal. Suppose that $f \in \bigcap _{n \in \mathbf{N}} I^ n$. Then Lemma 10.51.5 says that for every prime ideal $I \subset \mathfrak p$ there exists a $g \in R$, $g \not\in \mathfrak p$ such that $f$ maps to zero in $R_ g$. In algebraic geometry we express this by saying that “$f$ is zero in an open neighbourhood of the closed set $V(I)$ of $\mathop{\mathrm{Spec}}(R)$”.

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