I want to prove that: If $N$ is finitely generated semi-simple $R-$module, then $N$ is a sum of finitely many simple submodules.
I know that if $N$ is a finitely generated $R-$module, then that the following three conditions are equivalent(I have seen its proof before).
-
$N$ is a sum of simple modules.
-
$N$ is a direct sum of simple modules.
-
Given any submodule $M \subset N,$ there exists a unique submodule $M' \subset N$ such that $N = M \oplus M'.$
And any module satisfying these conditions is called a semi-simple module.
Then, to prove my first statement above, I know that I can adjust my proofs either for $2 \implies 1$ in the statement above or for $3 \implies 1.$ I prefer to prove my first statement to prove that: $N$ is a direct sum of simple modules implies $N$ is a sum of finitely many simple modules.
Here is my trial:
Assume that $N$ is a direct sum of simple modules, then, by definition of direct sum, $N$ is a sum of simple modules. But then how can I prove that it is a finite sum? could anyone help me in proving this please?
EDIT:
Maybe proving that every sequence of submodules of $N$ is stationary is a better way of proving my first statement but in that case I also do not see the proof of finitely many.
Any help is appreciated!
EDIT:
Since $N$ is semi-simple by assumption, then $N$ is a direct sum of (possibly infinitely many) simple modules. This infinitely many sum can come from either one of those two cases:
$(1)$ $N$ is infinitely generated i.e. generated by an infinite set. But this case is excluded because by assumption our $N$ is finitely generated.
$(2)$ at least one of the generators of $N$ is infinitely generated, which is not our case because each of the generators is simple i.e., it has only two submodules.
Is that reasoning correct or there is a more succinct proof?
Best Answer
A direct sum of infinitely many nonzero (right) modules is not finitely generated.
Indeed, if you're given a family of nonzero module $M_\lambda$, for $\lambda\in\Lambda$, and $x_1,\dots,x_n$ are elements of the direct sum thereof, there exists a finite set $F\subseteq\Lambda$ such that $$ x_1R+x_2R+\dots+x_nR\subseteq \bigoplus_{\lambda\in F}M_\lambda $$ and this finite direct sum is a proper submodule of the whole direct sum.