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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?

user26857
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calearner
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1 Answers1

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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.

user119882
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  • Why must that be so? Also, what is an example of such an infinite order element? – calearner Aug 15 '14 at 04:09
  • pick any nonzero x in P, then (x)=a product of some prime ideals, but (x) can not be a power of P (since P is of infinite order), so there exists a prime Q not P, such that x lies in Q. – user119882 Aug 16 '14 at 04:07