This is a quick question: given a finite group $G$, if we already know that for every $G$-module $M$, the space of invariants $M^G$ is a direct summand of $M$, how to show complete reducibility of $G$-modules? (this is from a note I'm reading)
And is this also true for representations of general associative algebra $A$? i.e. If we know that for any $A$-module $M$, $M^A$ is always a direct summand of $M$, is $A$ semisimple?