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.