Example of a module such that every proper submodule is finitely generated but the module is not.

Question

Let $R$ be a ring with 1 and $M$ an $R$-module. What is an example such that $M$ is infinitely generated but every proper submodule is finitely generated.

Answer 1

The direct limit of $\Bbb Z$-modules: $$\Bbb Z/p\to \Bbb Z/p^2\to \Bbb Z/p^3\to\cdots$$ is not finitely generated as a $\Bbb Z$ module but every proper submodule is isomorphic to $\Bbb Z/p^k$ for some $k$.

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