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

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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 ...

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Apologies in advance for this answer. The problem with it is that it is too advanced, and also relies on a lemma that is very similar to your question. I will continue to seek a more elementary answer. Lemma: Every left $R$ module over a left Artinian ring $R$ has a projective cover. Lemma: Every nonzero projective module has a maximal submodule. Lemma: ...

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