Let $K$ be a field. Let $A \subseteq B$ be an extension of Dedekind domains that are both finitely generated $K$-algebras. Assume that $A$ and $B$ have the same quotient field. Can we find an element $s \in A$ such that $B$ equals the localization $A_s$?
I would be very glad if this was a statement in some book, but a (not too complicated) proof would also be fine.
This is in general false. Take such an $A$, with $F$, the fraction field. Assume that there exists a prime ideal $P\subset A$ which is non-torsion in the Picard group. (These are plenty and happens more often than otherwise). Take $B$ to be the set of all $a\in F$ such that $aP^n\subset A$ for some $n$. Then $A\subset B$ and easy to see that $B\neq A_s$ for any $s\in A$.