# Are prime ideals of finite height in the powers series ring in infinitely variables finitely generated?

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.

• Actually my power series ring contains element s.t. X_1+ X_2+... , so it is strictly bigger than ∪ k[[X_1,...,X_n]]. In case of this union ring, the answer is true as in the argument by ACL. My question is whether the finite generatedness holds when the power series ring contains all infinite summation like X_1+ X_2+... Pierre Matsumi – Pierre MATSUMI Mar 9 '16 at 8:22