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.
Proof. Proof of (1). Let x_1, \ldots , x_ n be generators for the module N, see Lemma 10.51.1. For every prime \mathfrak p, I \subset \mathfrak p we see that the image of N in the localization M_{\mathfrak p} is zero, by Lemma 10.51.4. Hence we can find g_ i \in R, g_ i \not\in \mathfrak p such that x_ i maps to zero in N_{g_ i}. Thus N_{g_1g_2\ldots g_ n} = 0.
Part (2) follows from (1) and Lemma 10.23.1. \square
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