# Math Help - Prove the Artinian ring R is a division ring

1. ## Prove the Artinian ring R is a division ring

From a practice exam:

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

2. 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.

3. Ah, thanks!