Cardinality of fraction field

This is a ridiculous lemma that I came up with.

Lemma. Let be a (commutative) ring, and let be its total ring of fractions. Then and have the same cardinality.

Proof. If is finite, my previous post shows that . If is infinite, then is a subquotient of , hence . But injects into , so .

Corollary. If is a domain, then .

Proof. This is a special case of the lemma.

Finite domains are fields

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.