Let $R$ be a semi-simple ring, and let $M$ be an $R$-module. Show the following are equivalent:
(i) $M$ is finitely generated.
(ii) $M$ is noetherian.
(iii) $M$ is artinian.
I know the following:
Let $R$ be a semi-simple ring. Then i) $R$ is left-artinian and left-noetherian. ii) Every $R$-module is semi-simple.
Let R be a noetherian ring. Then an R-module is noetherian if and only if it is finitely generated.
So (i) $\iff$ (ii).
Let R be a artinian ring. Then an R-module is artinian if and only if it is finitely generated.
So (i) $\iff$ (iii).
Am I on the right tracks and what is left to prove?