Lemma 10.66.17. Let R be a ring. Let M be an R-module. The map
M \longrightarrow \prod \nolimits _{\mathfrak p \in \text{WeakAss}(M)} M_{\mathfrak p}
is injective.
Proof. Let x \in M be an element of the kernel of the map. Set N = Rx \subset M. If \mathfrak p is a weakly associated prime of N we see on the one hand that \mathfrak p \in \text{WeakAss}(M) (Lemma 10.66.4) and on the other hand that N_{\mathfrak p} \subset M_{\mathfrak p} is not zero. This contradiction shows that \text{WeakAss}(N) = \emptyset . Hence N = 0, i.e., x = 0 by Lemma 10.66.5. \square
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