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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.