Order (ring theory)

Summary

An order is a subring of a finite-dimensional algebra over a field. It is a finitely generated module over a ring of integers such as $\Z$, and it spans the algebra over the field. The basic examples are the subrings of an algebraic number field that are free of full rank over $\Z$, such as $\Z[\sqrt{-3}]$ in $\Q(\sqrt{-3})$. Orders in noncommutative algebras include $M_n(\Z)$ in $M_n(\Q)$ and the integral group ring $\Z G$ in $\Q G$. They are central in integral representation theory and in the arithmetic of quaternion algebras. The largest orders, the maximal orders, generalize rings of integers.