Prove the Artinian ring R is a division ring

• June 8th 2011, 03:03 AM
hatsoff
Prove the Artinian ring R is a division ring
From a practice exam:

Quote:

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!
• June 8th 2011, 03:35 AM
NonCommAlg
Quote:

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.
• June 8th 2011, 03:53 AM
hatsoff
Ah, thanks!