In mathematics, an order in the sense of ring theory is a subring \mathcal{O} of a ring A, such that The last two conditions can be stated in less formal terms: Additively, \mathcal{O} is a free abelian group generated by a basis for A over \mathbb{Q}. More generally for R an integral domain with fraction field K, an R-order in a finite-dimensional K-algebra A is a subring \mathcal{O} of A which is a full R-lattice; i.e. is a finite R-module with the property that . When A is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal orders exist in general, but need not be unique: there is in general no largest order, but a number of maximal orders. An important class of examples is that of integral group rings.

Examples

Some examples of orders are: A fundamental property of R-orders is that every element of an R-order is integral over R. If the integral closure S of R in A is an R-order then the integrality of every element of every R-order shows that S must be the unique maximal R-order in A. However S need not always be an R-order: indeed S need not even be a ring, and even if S is a ring (for example, when A is commutative) then S need not be an R-lattice.

Algebraic number theory

The leading example is the case where A is a number field K and \mathcal{O} is its ring of integers. In algebraic number theory there are examples for any K other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension ' of Gaussian rationals over \mathbb{Q}, the integral closure of \mathbb{Z} is the ring of Gaussian integers ' and so this is the unique maximal \mathbb{Z}-order: all other orders in A are contained in it. For example, we can take the subring of complex numbers of the form a+2bi, with a and b integers. The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.