I was reading about semisimple modules, and there's a fact relating to simple modules that I don't recall.
Suppose you have a ring $R=M_n(D)$, by which I mean the $n\times n$ ring of matrices with entries in a division ring $D$. Then actually, $R\cong M^n$ for some simple module $M$, and this $M$ is unique up to isomorphism.
I'm hoping to understand this before proceeding, but I don't see this isomorphism, nor why such a module is unique up to isomorphism. Can someone please explain? A kind thanks all.