Modules with finite support in $\mathrm{Max}(R)$

Question

Let $R$ be a commutative Noetherian ring, and $M$ be an $R$-module. Is the following statement true?

If $\mathrm{Supp}_R(M)$ (support of $M$) is a finite subset of $\mathrm{Max}(R)$ (the set of all maximal ideals of $R$), then $M$ is an Artinian $R$-module?

Thanks.

Answer 1

What about the $\mathbb Z$-module $\oplus_{n\ge1}\mathbb Z/2^n\mathbb Z$?

Answer 2

Well this is certainly wrong as long as you don't assume M to be finitely generated (just take an infinite dimensional vector space).

If M is finitely generated this should be true, even without the assumption that supp(M) is finite (which will rather be a consequence).

First note that this is obviously true if R is artinian,since M is a quotient of some $R^{n}$. Now let R be any noetherian Ring. The support of M is given by the prime ideals which contain Ann(M). Also M can be given the structure of an $R/Ann(M)$-Module. But R/Ann(M) is artinian since any prime ideal is maximal, so we can reduce to the artinian case.