Order (ring theory)


In mathematics, an order in the sense of ring theory is a subring of a ring, such that
  1. ' is a finite-dimensional algebra over the field of rational numbers
  2. spans ' over, and
  3. is a -lattice in '.
The last two conditions can be stated in less formal terms: Additively, is a free abelian group generated by a basis for ' over.
More generally for ' an integral domain contained in a field ', we define to be an '-order in a '-algebra ' if it is a subring of ' which is a full '-lattice.
When ' 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 '-orders is that every element of an '-order is integral over '.
If the integral closure ' of ' in ' is an '-order then this result shows that ' must be the maximal '-order in '. However this hypothesis is not always satisfied: indeed ' need not even be a ring, and even if ' is a ring then ' need not be an -lattice.

Algebraic number theory

The leading example is the case where ' is a number field ' and is its ring of integers. In algebraic number theory there are examples for any ' 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, the integral closure of ' is the ring of Gaussian integers ' and so this is the unique maximal '-order: all other orders in ' are contained in it. For example, we can take the subring of the complex numbers in the form, with and 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.