Let be a ring with . Prove that if is an ideal such that is a field, then is maximal in . (Do not assume that is commutative.)

Suppose we have a two-sided ideal with . By the Lattice Isomorphism Theorem for rings, is a two-sided ideal of the field . In particular, is commutative, and so by Proposition 9, the only ideals of are and . Again using the Lattice Isomorphism Theorem, we have or . So is maximal in .

## Comments

I think you accidentally assumed that $R$ is commutative. The use of Proposition 9 sort of depends on it. Any way to solve this without assuming such?

I think we’re only assuming that is commutative, which is true since we assumed is a field (and all fields are commutative in D&F). I rearranged the proof to (hopefully) make this more clear. Of course it is possible that I am completely confused.

Thanks!