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 $(3) \Rightarrow (1)$.


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 $M$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.