Lemma 10.66.17. Let $R$ be a ring. Let $M$ be an $R$-module. The map

is injective.

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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