Primes in denominators of algebraic numbers

Denote the set of algebraic numbers as $\bar{\mathbb{Q}}$ and the set of algebraic integers as $\bar{\mathbb{Z}}$. For $\gamma \in \bar{\mathbb{Q}}$, consider its irreducible polynomial in $\mathbb{Z}[x]$, $F_{\gamma }(x)=a_n{x}^n+\cdots +a_0$. Denote $e(\gamma )=\gcd (a_n,a_{n-1},\dots ,a_1)$. Drungilas, Dubickas and Jankauskas show in a recent paper that $\mathbb{Z}[\gamma ]\cap \mathbb{Q}=\{\alpha \in \mathbb{Q}\mid \{p\mid v_p(\alpha )<0\}\subseteq \{p\mid p\mathrm{\mid }e(\gamma )\}\}$. Given a number field $K$ and $\gamma \in \bar{\mathbb{Q}}$, 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 $\gamma \in \bar{\mathbb{Q}}$, we can find a finite set $S\subseteq \bar{\mathbb{Z}}$, such that for every number field $K$, we have $X(K,\gamma )=\{\mathfrak{p}\in \text{Spec}({\mathcal{O}}_K)\mid \mathfrak{p}\cap S\mathrm{\ne }\mathrm{\varnothing }\}$. We study how this set $S$ relates to the ring $\bar{\mathbb{Z}}[\gamma ]$ and the ideal ${\mathfrak{D}}_{\gamma }=\{a\in \bar{\mathbb{Z}}\mid a\gamma \in \bar{\mathbb{Z}}\}$ of $\bar{\mathbb{Z}}$. We also show that ${\gamma }_1,{\gamma }_2\in \bar{\mathbb{Q}}$ 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 $K$.