From a practice exam:

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.Quote:

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

Thanks!