This is a example in: Frank W. Anderson, Kent R. Fuller (auth.) Rings and Categories of Modules
$D$ be a division ring, not a field. It's mean $D$ is non-commutative. $C_n(D)$ and $R_n(D)$ satisfy eight axioms of vector space. So what's "right" and "left" vector space above meaning?
I can't prove that $C_n(D)$ is simple left $M_n(D)$-module and $R_n(D)$ is simple right $M_n(D)$-module.
What's "primitive diagonal idempotents"? I know "primitive idempotents".
Thanks for regarding!