Thread: Prove the Artinian ring R is a division ring

From a practice exam:

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.

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!

Originally Posted by hatsoff

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.

Thanks!

it's a trivial result of Artin-Wedderburn theorem.

Ah, thanks!