Category R-MOD, Modules finitely generated

Question

I'm trying to crack this one but it is harder than I thought, any help is welcome !

Let $M$ be a Semi-simple module. Show that $L(M) \in \mathbb{N} \iff $ $M$ is finitely generated.

I've proven $\rightarrow$]. I'm stucked on the other direction.

Answer 1

If I understand correctly, you want to show that a semisimple module is finitely generated if and only if it has finite length.

Let $M$ be semisimple, so we can write $M=\bigoplus_{i\in I}S_i$ as a direct sum of simple modules. Note that the length of $M$ is the cardinality of $I$, and that every simple is cyclic, so finitely generated.

Now, if $I$ is finite, then only finitely many simples occur, and taking one generator for each of them, we see that $M$ is finitely generated.

If $M$ is finitely generated, then each generator lies in only finitely many of the simples $S_i$, so all the generators lie in some finite direct sum, and since they generate we must have that $I$ is finite.