Simple quotient of a semisimple module is a direct summand?

Question

I am particurarly referring to Zimmermann's Representation Theory Corollary $1.4.20$.

Let $K$ be a field and $A$ a finite dimensional semisimple $K$-algebra. We have the decomposition of the regular $A$-module $$A = \oplus S_i$$ in the direct sum of simple $A$-modules. He wants to prove that if $S$ is a simple $A$-module, then $S \simeq S_i$ for some $i$.

In order to do so, he first proves that $S$ is a quotient of $A$ via the epimorphism $a\mapsto a\cdot s$, where $s\in S-\{0\}$. Then he concludes by saying that since the regular module $A$ is semisimple, this means that $S\simeq S_i$ for some $i$, but I can't see why.

How does he infer from the semi-simplicity of $A$ that the quotient is isomorphic to a direct summand?

Answer 1

So, we already know that $A=\oplus_iS_i$.

First, consider the mentioned map $\varphi:a\mapsto a\cdot s$ with an arbitrary nonzero $s$ from the simple $A$-module $S$.
This is indeed surjective as its image is a nontrivial submodule of $S$.

Now consider $\varphi_i:S_i\hookrightarrow \oplus_i S_i\cong A\to S$.
Since $\oplus_i\varphi_i=\varphi$, not all of these are the zero map, i.e. there is at least one index $i$, such that $\ker\varphi_i\ne S_i$.

By simplicity of $S_i$, then $\ker\varphi_i=0$, i.e. $\varphi_i$ is injective, and is also surjective by simplicity of $S$.