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The support of a module is closed?
Is there a simple example of a module $M$ of a Noetherian commutative ring $R$ such that $\operatorname{Supp}(M)\subset\operatorname{Spec}(R)$ is not closed?
When typing this question, this answer popped up.
So I think we can take $R=\mathbb{Z}$, and let $M=\bigoplus_{\mathfrak{p}\in S}\mathbb{Z}/\mathfrak{p}$ for a nonclosed subset $S$ of $\operatorname{Spec}(\mathbb{Z})$.
Is there an actual explanation as to why the support of such $M$ is not closed in $\operatorname{Spec}(\mathbb{Z})$? I didn't gather one from the original answer.
(I don't mind seeing a completely different example either, I just figured I'd ask about this one since it's already here.)