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 .