This is one of the classics.
Lemma. Let be a finite commutative ring. Then every element is either a unit or a zero-divisor.
Proof. If is not a zero-divisor, then the map
is injective. Since
is finite, it is also surjective, so there exists
with
.
Corollary 1. Let be a finite commutative ring. Then
is its own total ring of fractions.
Proof. The total ring of fractions is the ring , where
is the set of non-zerodivisors. But that set consists of units by the lemma above, so inverting them doesn’t change
.
Corollary 2. Let be a finite domain. Then
is a field.
Proof. In this case, the total ring of fractions is the fraction field. Therefore, is its own fraction field by Corollary 1.