Let k be any field, then how does one find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$? I believe there are two.
If we consider a rational function field $k(x,y)/k(x)$, then the only valuation ring which doesn't contain $k[x,y]$ is $k(x)[z]_{(z)}$ where $z=1/y$.
Notice that $k(x,y)/k(xy)$ is also a rational function field because $k(xy,x)=k(x,y)$. So shouldn't there only be one such valuation ring (based on what is known about rational function fields)?
(EDIT 14Aug2014) ANSWER: The only two valuation rings of the field extension $k(x,y)/k(xy)$ that do not contain $k[x,y]$ are $k(xy)[x^{-1}]_{(x^{-1})}$ and $k(xy)[y^{-1}]_{(y^{-1})}$.