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



Is the support of an Artinian module finite?
The answer is **yes**.


You can find a proof in Leamer's thesis [Homology of Artinian Modules Over Commutative Noetherian Rings](http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1020&context=mathstudent&sei-redir=1#search=%22artinian%20module%20noetherian%20rings%22), which contains the the following more precise result (see Lemma 3.1.11): 



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> Let $R$ be a commutative noetherian ring and let $M$ be an artinian $R$-module. Then
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> 1) the support of $M$ consists entirely of maximal ideals of $R$, that is, $\textrm{Supp}(M) \subseteq \textrm{m-Spec}(R)$;
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> 2) we have $\textrm{Min}\_R(M)=\textrm{Ass}\_R(M)=\textrm{Supp}\_R(M)$;
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> 3) $\textrm{Supp}\_R(M)$ is finite.
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