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.