Let $S$ be the local ring of nodal curve, $R=\varinjlim_{\text{Frob}} S \to S$. For example:
- $k$ a perfect field,
- $f(x,y)=y^{p+1}-x^{p+1}(1+x)$,
- $R=k[x^{1/p^{\infty}},y^{1/p^{\infty}}]/(f^{1/p^{\infty}})$.
Here's a complete local example:
- $k$ a perfect field,
- $f(x,y)=y^{p}-x^{p}y-x^{p+1}$,
- $R=k[[x^{1/p^{\infty}},y^{1/p^{\infty}}]]/(f^{1/p^{\infty}})$.
In each example $(y/x)$ is integral over $R$.
