Let $k$ be a field. Consider the inclusion of $k[x, xy, xy^2 - y]$ into $k[x,y]$. I claim that this is an epimorphism. Note that it is an inclusion, no non-units become units, and $k[x,y]$ has no idempotents.
Suppose $f$ and $g$ are two morphisms from $k[x,y]$ to some other commutative ring which agree on the given subring. Using $f(xy)=g(xy)$ and $f(x)=g(x)$, we see that $f(xy^2)=g(xy^2)$:
What does it mean when we say $f$ and $g$ agree on the given subring?
$f $ and $g $ agree on some subring means that for any element $x $ in the given subring, we have $f (x)=g (x). $