Let $\iota \colon A \subset B$ be a finite integral extension between domains. Suppose that $A$ is UFD, so $A$ is an integrally closed domain.
$A$ and $B$ may not be noetherian ring.

Choose a prime ideal ${\frak p}$ of $A$ such that ${\mathrm{ht}}(\frak p) < \infty$. By "lying-over" theorem, there are finitely many prime ideals
${\frak P}_1,\ldots,{\frak P}_n$ of $B$ such that ${\frak P}_i \cap A = {\frak p}$ for $1 \leq i \leq n$.

## Q. Does the following equality holds$\colon$ $\sqrt{{\frak p}B} = {\frak P}_1 \cap {\frak P}_2 \cap \ldots \cap {\frak P}_n$?