Prove the Artinian ring R is a division ring

From a practice exam:

Quote:

Let $\displaystyle R$ be a ring with identity $\displaystyle 1$, and assume that $\displaystyle R$ is right Artinian. Prove that if $\displaystyle R$ has no nonzero nilpotent ideals and no idempotent elements other than $\displaystyle 0$ and $\displaystyle 1$, then $\displaystyle R$ is a division ring.

I thought maybe we could use Schur's lemma for this one, but I don't see how to make it work with or without that result. Any help would be much appreciated.

Thanks!