Isaacs, $\textit{Character Theory of Finite Groups}$, Theorem(1.10)
Let $V$ be an $A$-module. Then $V$ is completely reducible iff it is a sum of irreducible submodules.

Pf: $\Leftarrow$ 

• $V=\sum V_{\alpha}$ and $W\subseteq V$
• $U\subseteq V$ maximal such that $W\cap U=0$ 

 $\Rightarrow$

• Let $S$ be the sum of all the irreducible submodules of $V$