is a field iff it has no nonzero proper ideals. So if is a field, . Conversely, if then is a maximal ideal (as Assuming choice, commutative ring has a maximal ideal and A maximal ideal in a commutative ring is prime), and thus has no nonzero proper ideals: is a field.

Note for every nonzero prime ideal is maximal by vacuity. Given , then any nonzero prime ideal is contained within a maximal ideal which is also prime (since Every ideal in a commutative ring is contained in a maximal ideal), but this must be equal to or else implies .