Order in a quaternion algebra

Let $B$ be a quaternion algebra over a number field $K$ . An order in $B$ is a subring $\mathcal O\subseteq B$ containing $1$ and $\mathcal O_K$ , where $\mathcal O_K$ is the ring of integers, such that $\mathcal O$ is a finitely generated $\mathcal O_K$ -module and spans $B$ over $K$ .

Example

In the split algebra $M_2(K)$ , the subring $M_2(\mathcal O_K)$ is an order: the four matrix units give an integral module basis and span all matrices over $K$ .

Finiteness is not freeness

Over a general number ring, a full finitely generated module need not be free. Thus “has an integral basis of four elements over $\mathcal O_K$ ” would be an unnecessarily restrictive definition.

An order is maximal when it is contained in no larger order. Maximality is not required in the definition of an arithmetic Kleinian group.

References

F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, Arithmeticity, discreteness and volume, Transactions of the AMS 349 (1997). Author-hosted paper, §4, integral orders used in the arithmetic-group construction.