Height one prime avoidance in normal domains

Question

Let $R$ be a Noetherian normal domain. Let $X$ be the set of height one prime ideals of $R$, and let $\mathfrak p \in X$. Can one have $$ \mathfrak p \subseteq \bigcup_{\mathfrak q \in X \setminus \{\mathfrak p\}} \mathfrak q? $$ Moreover, if this is impossible in a Noetherian normal domain, can it happen in a Krull domain?

Answer 1

Let $A$ be a Dedekind domain such that there exists a prime ideal $\mathfrak{p}$ of infinite order in the class group. Then $\mathfrak{p}$ is contained in the union of other prime ideals.