TITLE: Is the support of an Artinian module finite?
QUESTION [11 upvotes]: Let $R$ be a commutative Noetherian ring, $M$ is an Artinian $R$-module. Is the set $Supp_R(M)$ finite?
Thanks.

REPLY [6 votes]: It is well-known that for finitely generated Artinian $M$, the support is a finite set of maximal ideals. Since $\mathrm{Supp}(M)=\bigcup_{M'\subseteq M\ \mathrm{f.g.}}\mathrm{Supp}(M')$, we know that $\mathrm{Supp}(M)$ consists of maximal ideals. For any finite set $S\subseteq\mathrm{Supp}(M)$, it again follows easily from the finitely-generated case that $f_S\colon M\to\prod_{s\in S}M_s$ is surjective. In particular, for a strictly increasing chain $S_1\subset S_2\subset\dots$ of finite subsets of $\mathrm{Supp}(M)$, the chain of submodules $\ker(f_{S_i})$ is strictly decreasing, so $\mathrm{Supp}(M)$ must be finite.