Equivalence of Semisimplicity

Question

In Noncommutative algebra by Benson Farb, there is an exercise concerning this result;

An R-module M is semisimple if every submodule of M is a direct summand.

where semisimple is defined as M being the direct sum of simple R-submodules. I have finished all but part of the proof, in which I need to prove that,

Finitely generated submodules are noetherian.

So far, I have been trying to use the fact that every submodule 'inherits' the property that all submodules are direct summands. But I have no clue as to how a can prove this latter statement, after trying to use all of the equivalent definitions of Noetherian I know.

Answer 1

I think this is what you're asking:

Show that a finitely generated module $M$ in which every submodule is a direct summand is Noetherian.

Why not use the equivalent formulation: "$M$ is Noetherian iff every submodule is finitely generated."?

Suppose $N$ is a submodule of $M$. Then $N\oplus N'=M$ for some $N'$. We've been given that $M$ is finitely generated and $N$ is a quotient of $M$, so...