ac.commutative algebra - Do all Dedekind domains have the "Riemann-Roch property"?

Let $R$ be a Dedekind domain with fraction field $K$.

Say that a Dedekind domain $R$ has the Riemann-Roch property if: for every nonzero prime ideal $mathfrak{p}$ of $R$, there exists an element $f in (bigcap_{mathfrak{q} neq mathfrak{p}} R_{mathfrak{q}}) setminus R$, i.e., an element of $K$ which is integral at every prime ideal $mathfrak{q} neq mathfrak{p}$ and is not integral at $mathfrak{p}$.

Do all Dedekind domains have the Riemann-Roch property?

Motivation: for any subset $Sigma subset operatorname{MaxSpec}(R)$, put $R_{Sigma} := bigcap_{mathfrak{p} in Sigma} R_{mathfrak{p}}$. Then the maximal ideals of $R_{Sigma}$ correspond bijectively to the maximal ideals $mathfrak{p}$ of $R$ such that $mathfrak{p} R_{Sigma} subsetneq R_{Sigma}$. Thus $operatorname{MaxSpec}(R_{Sigma})$ may be viewed as containing $Sigma$. $R$ has the Riemann-Roch property iff for all $Sigma$,
$operatorname{MaxSpec}(R_{Sigma}) = Sigma$. Equivalently, the mapping $Sigma mapsto R_{Sigma}$ is an injection.

Remarks: $R$ has the Riemann-Roch property if its class group is torsion: then for every $mathfrak{p} in operatorname{MaxSpec}(R)$ there exists $n in mathbb{Z}^+$ and $x in R$ such that $mathfrak{p}^n = (x)$, so take $f = frac{1}{x}$. Also the coordinate ring $k[C]$ of a nonsingular, integral affine curve $C$ over a field $k$ has the Riemann-Roch property...by the Riemann-Roch theorem. Unfortunately this already exhausts the most familiar examples of Dedekind domains!