Let $D$ be a skew field that is central and finite-dimensional over a number field $F$ (in particular: a quaternion algebra over $F$). Let $\Delta \subseteq D$ be a maximal $\mathcal{O}_{F}$-order. Let $\mathfrak{b}$ be a fractional left $\Delta$-ideal and say we have a $\mathbb{Z}$-basis $\omega_1, \dotsc , \omega_n$ for $\Delta$ with $\omega_1 = 1$.

According to the definition of fractional ideals, we can find $d \in \Delta$ such that $d\mathfrak{b}$ (or $\mathfrak{b}d$?) $\subseteq \Delta$.

Given this setting, what do we know about $d$? A paper I have recently read has led me to believe that $d \in \mathbb{Z}$ but I don't see why that would be the case.