# Why is the cardinality of the codomain of a ring epimorphism at most the cardinality of the domain?

According to this page and thence linked text, if $e : R \to S$ is an epimorphism of rings, then the cardinality of $S$ cannot exceed the cardinality of $R$. This is a non-trivial observation because epimorphisms of rings need not be surjective. Is there a "layman's" explanation of this fact, one that does not require me to learn French?

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If I understand the link to the Stacks Project blog correctly, they rewrote this here : math.columbia.edu/algebraic_geometry/stacks-git/… – Matthieu Romagny Nov 1 '11 at 20:54
So...if $R$ if is an integral domain and $S$ is its field of fractions, then $e$ is automatically an isomorphism. That gives an interesting proof that a finite integral domain is a field. Or, maybe, the statement in the question should be regarded as a generalization of this fact. – George Lowther Nov 1 '11 at 20:57
@George Lowther : I don't think you are right. Indeed, the injection of an integral domain into its field of fractions is always a ring epimorphism (direct check). – Matthieu Romagny Nov 1 '11 at 21:22
@Matthieu: Yes, that's what I was getting at. It is always an epimorphism and, if it is an integral domain, it is one-to-one. If $R$ is also finite then $|R|\ge|S|$ is equivalent to e being onto (hence, an isomorphism). – George Lowther Nov 1 '11 at 22:33
George, I agree if R is finite, but that assumption was neither in the original question nor in your first comment. In other words : you should really write "if R is a finite integral domain etc". Best, – Matthieu Romagny Nov 2 '11 at 13:12

An explanation of a layman to a layman. Let $T={\rm Im}\ e$. Then embedding $T\to S$ is again an epimorphism. With respect to $T$ the ring $S$ behaves like the ring (not necessarily the field!) of fractions (compare $\mathbb{Z}$ and $\mathbb{Q}$), so it has the same cardinality. It is enough? :-) I think it is possible to give a rigorous proof by so-called "zigzag-theorem" from Theory of semigroups.
What do you mean by "behaves like" a ring of fractions? That every elemente of $S$ is a quotient of elements from $T$? – Andrej Bauer Nov 2 '11 at 8:45