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Claim: Every non-trivial finitely generated $R$-module has a non-trivial simple quotient.

How to prove this? I was able to show that a $R$-module is simple if and only if it is isomorphic to a module of the form $R/m$ where $m$ is maximal. Is this helpful to prove the above claim?

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This is the same as proving that such a module $M$ has a maximal non-trivial submodule $N$. You can prove this by a Zorn's lemma argument in much the same way as one proves a ring has a maximal ideal. The finite generation of $M$ is important.

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