TITLE: The intersection of an infinite number of prime ideals in a ring of integers
QUESTION [9 upvotes]: Let $\mathcal{O}$ be the ring of integers of a number field, $\{\mathfrak{p}_i,\,i \in \mathbb{N}\}$ a sequence of two-by-two pairwise distinct prime ideals. Does it follow that$$\bigcap_i \mathfrak{p}_i = \{0\}?$$

REPLY [2 votes]: The intersection of ideals is an ideal. Since yours is divisible by infinitely many primes, it must be that it has no presentation as a product of primes. But in a Dedekind domain, every non-zero ideal has a unique expression as the finite product of prime ideals (to some power). Hence it must be that your ideal is the only ideal not covered by this theorem:  the zero ideal.