counterexample that $M$ is not finitely generated $R$-module and $M$ has no maximal submodule

Question

In the proof of Nakayama Lemma, the following proposition is used:

let $R$ be a commutative ring with identity and $M$ be a non-zero finitely generated $R$-module. Then $M$ has a maximal submodule.

Is there any counterexample that $M$ is not finitely generated $R$-module and $M$ has no maximal submodule? Thanks.

Answer 1

Consider $\mathbb Z_{p^\infty}$ as $\mathbb Z$-module. It is not hard to see that $\mathbb Z_{p^\infty}$ is not finitely generated $\mathbb Z$-module. Obviously $\mathbb Z_{p^\infty}$ has no maximal subgroup.