Ideal class set of a quaternion order

Let $\mathcal O$ be an order in a quaternion algebra $B$ over a number field. Write $\mathcal I_R(\mathcal O)$ for its locally principal right fractional ideals. Its right ideal class set is

\operatorname{Cls}_R(\mathcal O)=\mathcal I_R(\mathcal O)/\!\sim, \qquad I\sim J\iff J=aI\text{ for some }a\in B^\times.

The ideals are locally principal relative to the specified right order. The cardinality $h(\mathcal O)=\#\operatorname{Cls}_R(\mathcal O)$ is its right class number; it is finite in this number-field setting.

Why the scalar acts on the left

Left multiplication by $a$ gives an isomorphism of right $\mathcal O$ -modules $I\to aI$ . Conversely such a module isomorphism extends over the fraction field to an isomorphism of right $B$ -modules, which must be left multiplication by an element of $B^\times$ .

A pointed set, not generally a group

The class of $\mathcal O$ is distinguished, but ordinary ideal multiplication does not generally define a group law on these classes. Noncommutative left and right orders must be compatible before multiplying ideal classes. This differs from the commutative Picard group.

The Hurwitz order has class number one. Its type number also equals one, but type number counts orders rather than ideals.

References

John Voight, Quaternion Algebras, author's text, Definitions 17.3.1 and 17.3.4, Remarks 17.3.5–17.3.6, Main Theorem 17.7.1, and §11.3 for the Hurwitz order.