Primes in denominators of algebraic numbers

Denote the set of algebraic numbers as $\overline{\mathbb{Q}}$ and the set of algebraic integers as $\overline{\mathbb{Z}}$. For $\gamma\in\overline{\mathbb{Q}}$, consider its irreducible polynomial in $\mathbb{Z}[x]$, $F_{\gamma}(x)=a_nx^n+\dots+a_0$. Denote $e(\gamma)=\gcd(a_{n},a_{n-1},\dots,a_1)$. Drungilas, Dubickas and Jankauskas show in a recent paper that $\mathbb{Z}[\gamma]\cap \mathbb{Q}=\{\alpha\in\mathbb{Q}\mid \{p\mid v_p(\alpha)<0\}\subseteq \{p\mid p|e(\gamma)\}\}$. Given a number field $K$ and $\gamma\in\overline{\mathbb{Q}}$, we show that there is a subset $X(K,\gamma)\subseteq \text{Spec}(\mathcal{O}_K)$, for which $\mathcal{O}_K[\gamma]\cap K=\{\alpha\in K\mid \{\mathfrak{p}\mid v_{\mathfrak{p}}(\alpha)<0\}\subseteq X(K,\gamma)\}$. We prove that $\mathcal{O}_K[\gamma]\cap K$ is a principal ideal domain if and only if the primes in $X(K,\gamma)$ generate the class group of $\mathcal{O}_K$. We show that given $\gamma\in \overline{\mathbb{Q}}$, we can find a finite set $S\subseteq \overline{\mathbb{Z}}$, such that for every number field $K$, we have $X(K,\gamma)=\{\mathfrak{p}\in\text{Spec}(\mathcal{O}_K)\mid \mathfrak{p}\cap S\neq \emptyset\}$. We study how this set $S$ relates to the ring $\overline{\mathbb{Z}}[\gamma]$ and the ideal $\mathfrak{D}_{\gamma}=\{a\in\overline{\mathbb{Z}}\mid a\gamma\in\overline{\mathbb{Z}}\}$ of $\overline{\mathbb{Z}}$. We also show that $\gamma_1,\gamma_2\in \overline{\mathbb{Q}}$ satisfy $\mathfrak{D}_{\gamma_1}=\mathfrak{D}_{\gamma_2}$ if and only if $X(K,\gamma_1)=X(K,\gamma_2)$ for all number fields $K$.