# Prove the Artinian ring R is a division ring

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