Lemma 10.66.8. Let $R$ be a ring. Let $M$ be an $R$-module. Any $\mathfrak p \in \text{Supp}(M)$ which is minimal among the elements of $\text{Supp}(M)$ is an element of $\text{WeakAss}(M)$.

**Proof.**
Note that $\text{Supp}(M_{\mathfrak p}) = \{ \mathfrak pR_{\mathfrak p}\} $ in $\mathop{\mathrm{Spec}}(R_{\mathfrak p})$. In particular $M_{\mathfrak p}$ is nonzero, and hence $\text{WeakAss}(M_{\mathfrak p}) \not= \emptyset $ by Lemma 10.66.5. Since $\text{WeakAss}(M_{\mathfrak p}) \subset \text{Supp}(M_{\mathfrak p})$ by Lemma 10.66.6 we conclude that $\text{WeakAss}(M_{\mathfrak p}) = \{ \mathfrak pR_{\mathfrak p}\} $, whence $\mathfrak p \in \text{WeakAss}(M)$ by Lemma 10.66.2.
$\square$

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