Let $k$ be a field of characteristic zero.
Is $k[x,xy] \subseteq k[x,y]$ a flat ring extension?
I guess that the answer is no? Though I am not sure how to prove this. Perhaps applying this criterion (showing it is not satisfied?) will help.
Any comments and hints are welcome!
Set $A=k[x,xy]$ and $B=k[x,y]$. (Notice that $A$ is isomorphic to a polynomial ring over $k$ in two variables.)
We have an exact sequence $0\to A/xA\stackrel{xy\cdot}\to A/xA$.
Assuming that $B$ is $A$-flat and tensoring by this we get that $0\to B/xB\stackrel{xy\cdot}\to B/xB$ is exact. But this is the zero map, so $B=xB$, a contradiction.