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.



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