Let $A$ be a UFD domain, i.e. integral and for any height one prime
${\frak p}$ of $A$, we have ${\frak p} = (u_{\frak p})$ for some $u_{\frak p} \in A$.

Once and for all, we fix the algebraic closure $\overline{K}$ and consider the integral closure $\overline{A}$ of $A$ in $\overline{K}$.

We consider, for a finite integer $d < \infty$, the following homomorphism of $A$-algebras $\colon$

\begin{equation}\label{P}
f \colon A[X_1,\ldots,X_d] \to \overline{A},
\end{equation}
where we define $X_i \mapsto a_i \in \overline{A}$.

We set ${\frak P} \colon= \mathrm{Kernel}(f)$, which is a prime ideal of $A[X_1,\ldots,X_d]$.

## Q. Even if $A$ is $not$ noetherian, is ${\frak P}$ always finitely generated ?