Are all simple left modules over a simple left artinian ring isomorphic?

I've a basic question.

$R$ is a simple left artinian ring. I want to show that all simple left $R$-modules are isomorphic.

A simple $R$-module $M$ is isomorphic to $R/J$ where $J$ is a maximal left ideal of $R$. Does this imply that $M\cong Ra$ for some $a\in R$, $a\neq 0$?

Many thanks.

Solution 1:

Yes: if $S$ is a simple left module, $S$ is isomorphic to $R/M$ for a maximal left ideal $M$. Then $R\cong M\oplus S'$ for some left ideal $S'$ since $R$ is semisimple, and $S\cong S'$. So it suffices to look only at minimal left ideals. (Incidentally, $S'$ has to be of the form $Re$ for an idempotent $e$, but we won't need that.)

Since $R$ is simple, it's prime, and so if $S$ and $T$ are nonisomorphic minimal left ideals, $ST\neq\{0\}$. But then $St=T$ for some $t\in T$, which provides an isomorphism of S with T : right multiplication by t.