I'm reading An introduction to homological algebra of Rotman, but the proposition 4.5 of the section 4.1 Semisimple rings states this:
The following conditions on a ring $R$ are equivalent.
$R$ is semisimple.
Every left (or right) $R$-module $M$ is a semisimple module.
Every left (or right) $R$-module $M$ is injective.
Every short exact sequence of left (or right) $R$-modules splits.
Every left (or right) $R$-module $M$ is projective.
And the proof of the first point to the second doesn't look very clear. This is the proof the book has:
Since $R$ is semisimple, it is semisimple as a module over itself;
hence, every free left $R$-module is a semisimple module. Now $M$ is a
quotient of a free module, by Theorem $2.35$, and so Corollary $4.2$ gives
$M$ semisimple.
I don't understand why the part in boldface is true. Can anyone explain to me the hence part?
Best Answer
By definition, a semisimple ring is a ring which is semisimple as a left module over itself. A free left $R$-module is a left module (isomorphic to a module) of the form $\bigoplus_{A} R$ for some index set $A$. Moreover, it is true that if $M_i$ is a collection of semisimple modules, then $\bigoplus_{i\in I}M_i$ is also semisimple. Putting all this together implies the result.