I'm reading Lang's Algebra on the section of semisimple ring. There is one proposition that says
If $R$ is semi simple, then every $R$-module is semi-simple.
Here we assume $R$ is unital but not necessarily commutative. In the proof, we are using that "every $R$-module is a quotient of a free module. My question is: do we need the finitely generated condition here?
Recall in Atiyah-Macdonalds, Proposition 2.3, we had that every finitely generated $A$-module $M$ is a quotient of a free module of rank $n$ where $n$ is finite.
This leads me to think that why did Atiyah-Macdonalds only restricted to the finitely generated case in commutative rings? Would there be an example where this fails for non-f.g. modules? Or is it true that this holds for all modules, no matter f.g. or not?