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.