Is there an example of a finite extension of Dedekind domains $R \subset S$ and a principal ideal $I \subset S$ such that $I \cap R$ is not a principal ideal of $R$ ?
I don't have good ideas for solving this. This question is a bit related, since any localization $R_p$ at a prime $p \subset R$ is a DVR.
Thank you.
On
Let $L$ be a number field of class number 1, and $K$ a subfield with nontrivial class group. Let $\mathfrak{p}$ be a non-principal prime in $K$ and $\mathfrak{P}$ a prime of $L$ lying over $\mathfrak{p}$. Then $\mathfrak{P}$ is principal, but $\mathfrak{P} \cap \mathcal{O}_K = \mathfrak{p}$ is not.
There's a theorem that says that every ideal of a field $K$ becomes principal in its Hilbert Class Field. So to get an example, we take $K=\mathbb{Q}(\sqrt{-5})$ and ideal $(2, 1+\sqrt{-5})$ which is nonprincipal, and look at it in its HCF which is $\mathbb{Q}(\sqrt{-5}, i)$. This ideal becomes the principal ideal $(1+i)$. This takes some work to check, obviously: the identities
$2=(1+i)(1-i)$
$1+\sqrt{-5}=(1+i)\left(\frac{1+\sqrt{5}}{2}-i \cdot\frac{1-\sqrt{5}}{2}\right)$
$1+i=\left(i\cdot\frac{1-\sqrt{5}}{2}\right)\cdot 2+(1+\sqrt{-5})$
show the two inclusions. So this is an example of the theorem, and an example of your request.