Let R $R$ be a commutative ring, let A $A$ be an R-algebra, $R$-algebra, and let M $M$ be an A-module. $A$-module. If M $M$ is simple, then End_{A-mod}(M) End$_{A-mod}(M)$ is a division ring.
A common use is when R $R$ is the complex numbers \CC, $\mathbb{C}$, and M $M$ is such that End_{A-mod}(M) End$_{A-mod}(M)$ is finite dimensional. Then End_{A-mod}(M) End$_{A-mod}(M) = \CC.mathbb{C}$.
Under what circumstances (regarding R $R$ and/or A) $A$) is the converse true, that the endormorphism endomorphism ring being a division ring, or being just R $R$ itself, implies that M $M$ is simple?
