Let $R$ be a unital ring, $M$ is a finitely generated $R$-module.
My question is to prove that there exist a maximal submodule in $M$. However I have no strategy to prove that except using the idea of Zorn lemma.
Can any body help me to solve this problem?
Also, please give a counter example for the case that if $M$ is not finitely generated.
Thank for reading. I beg your pardon for my poor English