I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \operatorname{frac}(R)$. He specialises to the case of a central simple algebra $A$ over a number field $K$, with $R=\mathcal{O}_K$, and mentions (page 15, section 5.2) that 'the set of conjugacy classes of maximal $R$-orders may not be singleton; its cardinality, if is finite, is called the type number of A.'

My question is: what do we know about type numbers? Does Jordan-Zassenhaus imply finiteness of the type number in this case? Any references on type numbers of central simple algebras would be greatly appreciated.

I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \operatorname{frac}(R)$. He specialises to the case of a central simple algebra $A$ over a number field $K$, with $R=\mathcal{O}_K$, and mentions (page 15, section 5.2) that 'the set of conjugacy classes of maximal $R$-orders may not be singleton; its cardinality, if is finite, is called the type number of $A$.'

My question is: what do we know about type numbers? Does Jordan-Zassenhaus imply finiteness of the type number in this case? Any references on type numbers of central simple algebras would be greatly appreciated.

I've been reading On the existence of maximal orders, by Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = frac(R)$. He specialises to the case of a central simple algebra $A$ over a number field $K$, with $R=\mathcal{O}_K$, and mentions (page 15, section 5.2) that 'the set of conjugacy classes of maximal $R$-orders may not be singleton; its cardinality, if is finite, is called the $\textit{type number}$ of A.'

My question is: what do we know about type numbers? Does Jordan-Zassenhaus imply finiteness of the type number in this case? Any references on type numbers of central simple algebras would be greatly appreciated.

I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \operatorname{frac}(R)$. He specialises to the case of a central simple algebra $A$ over a number field $K$, with $R=\mathcal{O}_K$, and mentions (page 15, section 5.2) that 'the set of conjugacy classes of maximal $R$-orders may not be singleton; its cardinality, if is finite, is called the type number of A.'

My question is: what do we know about type numbers? Does Jordan-Zassenhaus imply finiteness of the type number in this case? Any references on type numbers of central simple algebras would be greatly appreciated.