Let $M$ be an $A$-module. If every nonempty set of finitely generated submodules of $M$ has a maximal element, then $M$ is Noetherian.
One way to see this is that every submodule of $M$ is finitely generated submodule. Let $N$ be a finitely generated submodule, let $\Sigma$ be a set of all finitely generated submodules of $N$. Then $\Sigma$ has a maximal element, and if it is not $N$, then it contradicts its maximality.QED.
However, one suspicious thing is the assumption that $\Sigma$ is a set. Is $\Sigma$ really a set? In other words, a collection of all finitely generated submodules of a finitely generated module is a set?