Maximal order

Summary

A maximal order is an order in an algebra that is not contained in any larger order. In an algebraic number field the ring of integers is the unique maximal order, so maximal orders generalize rings of integers to noncommutative algebras. In noncommutative algebras maximal orders exist under mild hypotheses but are usually not unique. For example, the conjugates of $M_2(\Z)$ by elements of $GL_2(\Q)$ are maximal orders in $M_2(\Q)$. Maximal orders in quaternion algebras are central in the arithmetic of quaternions and of Shimura curves.