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Does there exist a Noetherian regular domain $R$ (necessarily non-local) and distinct maximal ideals $\mathfrak{m}_n$ for each $n \in \mathbb{N}$, such that $$\bigcap_n (\mathfrak{m}_n)^n \neq 0.$$ May be this is impossible by a clever application of Krull's intersection theorem, but I don't see why.

I am interested in the case where R has prime characteristic, but I suspect if an example exists it will be characteristic independent.

In prime characteristic, I can show that such examples are impossible for regular domains which are essentially of finite type over a Noetherian local ring $A$ whose formal fibers are geometrically regular (i.e. a $G$-ring) using techniques which seem overkill.

In low dimension, note that this is also cannot happen for a Dedekind domain because any non-zero element of a Dedekind domain is only contained in finitely many maximal ideals. I tried playing with some of the constructions of large Noetherian rings such as Nagata's Noetherian ring of infinite Krull dimension, but I think this ring also has the property that any element is contained only in finitely many maximal ideals based on the answer here: Noetherian ring with infinite Krull dimension (Nagata's example).

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