# The support of a module is closed?

It's well known that under some "weak" hypothesis, such as finitely generation, the support of an A-module is closed in Spec(A). It is true also in the most general case?

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Pick a non-closed subset $S$ of $\operatorname{Spec}\mathbb Z$, and let $\displaystyle M=\bigoplus_{\mathfrak p\in S}\:\mathbb Z/{\mathfrak p}$.
Very nice. I suppose by $\mathbb{Z}_p$ you mean the ring of $p$-adic integers. :) –  Martin Brandenburg Dec 9 '10 at 13:55
Do you mean $\mathbb{Z}/p$? The support of the localization $\mathbb{Z}_{(p)}$ is the entire space. –  Akhil Mathew Dec 9 '10 at 15:39