Question

Show that for a semisimple module M over any ring R, the following conditions are equivalent:
(1) M is finitely generated;
(2) M is noetherian;
(3) M is artinian;
(4) M is a finite direct sum of simple modules.

Answer 1

(1) $\Rightarrow$ (4). Let $M = \oplus_{i \in I} M_i$ where the $M_i$'s are simple modules. If $M$ is generated by $m1, \ldots, mn$, we have $\{m1, \ldots, mn\} \subseteq \oplus_{i \in J} M_i$ for a finite subset $J \subseteq I$. Therefore, $M = \oplus_{i \in J} M_i$ (which of course implies that $J = I$). (4) $\Rightarrow$ (2) $\Rightarrow$ (1) and (4) $\Rightarrow$ (3) are trivial, so we are done if we can show (3) $\Rightarrow$ (4). Let $M = \oplus_{i \in I} M_i$ as above. If $I$ is infinite, this decomposition of $M$ would lead to a strictly descending chain of submodules of $M$. Therefore, $I$ must be finite, and we have (4).