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Frobenius normalization of a prime ideal

For the prime ideal $P$ of a commutative Noetherian domain, is there a description of $\mathcal O_x$ for $x\in P$ which $x$ is an $\mathcal O_x$-regular element? In other words, if the Frobenius map is $x\to f(x)$, is there a description of $f(x)$?

If $\mathfrak{p}$ is a height $1$ prime of $A$ (in a Noetherian local ring $A$) then $A_\mathfrak{p}$ is a regular $A_\mathfrak{p}$-algebra and it is torsion-free as a left module over itself by the generalized Nakayama lemma; it is moreover an integral domain since $A$ is Noetherian. So a height $1$ prime $\mathfrak{p}$ is an integrally closed prime ideal.

I assume you mean that $P$ is prime in $A$. In this case I think the answer is no (though you never know). Let $x$ be an element in $P$ that is 0b46394aab

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