Let be a commutative ring with . Prove that the following are equivalent.
- has exactly one prime ideal.
- Every element of is either nilpotent or a unit.
- is a field.
Suppose has exactly one prime ideal. Since every maximal ideal of is prime, is local. By §7.4 #26, is the unique maximal ideal. By §7.4 #37, every element of is a unit, and (by definition) the remaining elements are nilpotent.
Suppose is nonzero; then . Since is not nilpotent in , is a unit. Then exists in , and we have . Thus every nonzero element of is a unit. Since this ring is commutative, is a field.
Suppose is a field. By §7.4 #26, is contained in every prime ideal of . By the Lattice Isomorphism Theorem for rings, the only possible proper prime ideal is ; moreover, is prime since it is maximal in . Thus is the unique prime ideal of .