Let $R$ be a commutative Noetherian ring, $M$ is an Artinian $R$module. Is the set $Supp_R(M)$ finite? Thanks.

The answer is yes. You can find a proof in Leamer's thesis Homology of Artinian Modules Over Commutative Noetherian Rings, which contains the the following more precise result (see Lemma 3.1.11):



It is wellknown 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 finitelygenerated 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. 

