Central idempotents of a ring which has only one simple module up to isomorphism

Let $R$ be a ring such that any two simple $R-modules$ are isomorphic. Show that $R$ has no nontrivial central idempotent.

I know how to prove this for a simple ring $R$, since a central idempotent produces a two-sided ideal. But why is it true for any ring in general?

If there is a nontrivial central idempotent $e$, then $R=eRe\oplus (1-e)R(1-e)$ is a decomposition into two rings. If $M$ is a maximal left ideal in $eRe$ and $N$ a maximal left ideal of $(1-e)R(1-e)$, then $I=M\oplus (1-e)R(1-e)$ and $J=eRe\oplus N$ are maximal left ideals of $R$.
Furthermore, $R/I$ and $R/J$ are nonisomorphic simple modules. (For example, $eRe$ annihilates $R/J$ but not $R/I$.