Let $A:= {\mathbb F}_p[[X_1,...,X_∞]]$ be the infinitely many variables formal power series ring over ${\mathbb F}_p$, which is UFD.

Consider an arbitrary prime ideal $P$ of $A$ such that the height of $P$ is finite.

## Question: Is P finitely generated?

If this is true, my previous question is solved in the affirmative. That is, for an arbitrary ideal $I$ of $A$ such that the height of $I$ is equal to $d$, there are only finitely many prime ideals $P_i$ such that $I \subseteq P_i$ and that the height of $P_i$ is equal to $d$. This follows from a Theorem by D.D.Anderson.

Actually, there is a Krull domain $R$ a certain height $1$ prime ideal $P$ of which is infinitely generated.