At first sight, my question seems quite arbitrary. I hope it is nevertheless interesting. Actually I wanted to know more about non-Noetherian domains having Krull dimension one and I was therefore searching for examples. One example of such a ring is the integral closure $A$ of $\mathbb{Z}$ in some algebraic closure $\bar{\mathbb{Q}}$ of $\mathbb{Q}$.
I already know that every non-zero prime ideal of $A$ contracts to a non-zero prime ideal of $\mathbb{Z}$ and that, given a prime $p \in \mathbb{Z}$, there are in general infinitely many prime ideals $M \subseteq A$ such that $M \cap \mathbb{Z} = (p)$. Let $V(p)$ denote the set of all such prime ideals of $A$.
Now to my actual question: Given some $Q \in V(p)$, can it happen that $\bigcap_{M \in V(p) \setminus \{Q\}} M \subseteq Q$, or is this never the case?
This would solve a problem I have been thinking about for quite a while.
Thank you very much in advance!