According to [this page](http://math.columbia.edu/~dejong/wordpress/?p=623) and thence [linked text](http://www.numdam.org/item?id=SAC_1967-1968__2__A2_0), 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?



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


