Let be a finite commutative ring with . Prove that every prime ideal of is maximal.
We begin with a lemma.
Lemma: If is a finite integral domain, then is a field. Proof: Let be nonzero, and define by . We claim that is injective. To see this, suppose ; then , so that . Since , , so that . Hence is injective. Since is finite, is also surjective. In particular, there exists such that , so that is a unit. Thus every nonzero element of is a unit, and is a field.
Now if is prime, then is a finite integral domain. Then is a field, and thus is maximal.