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In a (necessarily non-Noetherian) integral domain $A$ of (Krull) dimension $1$, is it possible that there is an infinite collection of prime ideals $\mathfrak{p}_i$ such that $\cap_i \mathfrak{p}_i \neq 0$?

Context: part of Problem 5.2.4 in Qing Liu's Algebraic Geometry and Arithmetic Curves asks you to show that (after reducing to the ring statement) that for an affine integral scheme $X \simeq \text{Spec } A$ of dimension $1$, an element $\frac{p}{q}$ of $K(X)$ is contained in $\mathcal{O}_{X,x}$ for all but finitely many $x$. This means $q\neq 0$ is contained in only finitely many prime ideals.

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Let $\mathcal{O}$ be the ring of all algebraic integers in $\overline{\Bbb{Q}}$. It is a non-Noetherian integral domain of dimension $1$. For any non-zero prime $\mathfrak{p} \in \Bbb{Z}$, there are infinitely many primes $\mathfrak{p}_i$ in $\mathcal{O}$ lying above $\mathfrak{p}$, and so $\bigcap \mathfrak{p}_i \neq 0$.

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