If $R$ is a local left artinian ring, and $A$ is a left artinian subring, is $A$ a local ring?
I can show that $rad A = A \cap rad R$ since $rad A$ and $rab R$ nilpotent ideals ($R$ and $A$ are artinian), and with the second isomorphism theorem, I can show that $A/rad A \cong (A + rad R) / rad R$. So I was hoping to show $(A + rad R) / rad R$ is a division ring, which would imply that $A$ is a local ring, but I'm not sure that I can show that. Any suggestions?
The following assumes $A$ and $R$ share identity.
If $A$ is right Artinian, the idempotents lift from $A/J(A)$ to $A$. If $A/J(A)$ has a nontrivial idempotent, so does $A$, but $R$ contains no nontrivial idempotents.
Then it can only be the case that $A/J(A)$ is a semisimple artinian ring with only trivial idempotents, and that amounts to a division ring.