Let $\mathcal{A}$ be a central simple algebra over an algebraic number field $K$, and $\mathcal{O}$ be a maximal $\mathcal{O}_K$-order in $\mathcal{A}$. Let $I$ be a maximal integral left-ideal of $\mathcal{O}$. Then there is a unique associated two-sided prime ideal of $\mathcal{O}$, say $\mathfrak{P}$, such that $\mathfrak{P}\subset I\subset\mathcal{O}$. Let $q\in\mathbb{Z}$ be a rational prime. My question is the following: are $\mathfrak{P}/q\mathfrak{P}$ and $I/qI$ isomorphic?
I know that the natural inclusion $\mathfrak{P}\hookrightarrow I$ induces an injection $\mathfrak{P}/q\mathfrak{P}\hookrightarrow I/qI$; I also know how to find the absolute norm of $\mathfrak{P}/q\mathfrak{P}$, which gives the cardinality of $|\mathfrak{P}/q\mathfrak{P}|$. However, I don't know how to find the cardinality of $I/qI$. In addition, I am also struggling to prove whether the map is surjective.