The exercise is as follows:
Show that for a semisimple module $M$ over any ring, the following conditions are equivalent:
$(1)$ $M$ is finitely generated;
$(2)$ $M$ is Noetherian;
$(3)$ $M$ is Artinian;
$(4)$ $M$ is finite direct sum of simple modules.
I managed to do the following implications: $(1) \Rightarrow (4)$, $(2) \Rightarrow (4)$, $(4) \Rightarrow (2)$ and $(4) \Rightarrow (3)$.
Thus leaving $(3) \Rightarrow (1)$.
Best Answer
If $M=\oplus_{i\in I} S_i$ is a direct sum of simple modules, then it's obvious that if $M$ is Artinian, $I$ has to be finite.
If you take one nonzero element from each of these simple modules, prove those elements generate $M$.