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. $\qedsymbol$

Lovely little lemmas

Or maybe ‘amusing observations’

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