Cardinality of fraction field

This is a ridiculous lemma that I came up with.

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

Proof. If R is finite, my previous post shows that R = K. If R is infinite, then K is a subquotient of R^2, hence |K| \leq |R^2| = |R|. But R injects into K, so |R| \leq |K|. \qedsymbol

Corollary. If R is a domain, then |\operatorname{Frac}(R)| = |R|.

Proof. This is a special case of the lemma. \qedsymbol

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