Let be a commutative ring with . Prove that is a field if and only if is a maximal ideal.
Suppose is a field. Let be an ideal which properly contains ; then there exists an element in . Since is a field, is a unit in . By Proposition 9, . Thus is the only ideal of which properly contains , and hence is a maximal ideal in .
Suppose is a maximal ideal in . Now is commutative by hypothesis. Let be nonzero. Then since is maximal, we have . Since , there exist elements such that . By §7.1 #28, has a two-sided inverse. Thus every nonzero element of is invertible, and hence is a field.