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 $R$.

Let $S$ be the local ring of nodal curve, $R$ = inverse limit $Frob: S \to S$. 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 $R$.

Let S=(local ring of) nodal curve, R=inverse limit Frobenius S->S. For example, k=perfect field, f(x,y)=y^p+1-x^p+1(1+x), R=k[x^1/p∞,y^1/p∞]/(f^1/p∞). Here's a complete local example: k=perfect field, f(x,y)=y^p-x^py-x^p+1, R=k[[x^1/p∞,y^1/p∞]]/(f^1/p∞). In each example (y/x) is integral over R.

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 $R$.