# Order (ring theory)

In mathematics, an **order** in the sense of ring theory is a subring of a ring , such that

*is a finite-dimensional algebra over the field of rational numbers*- spans
*over , and* - 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 with fraction field **, an **-order in a finite-dimensional **-algebra ** is a subring of ** which is a full **-lattice; i.e. is a finite **-module with the property that **.*^{[1]}

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[edit]

Some examples of orders are:^{[2]}

- If is the matrix ring over , then the matrix ring over is an -order in
- If is an integral domain and a finite separable extension of , then the integral closure of in is an -order in .
- If in is an integral element over , then the polynomial ring is an -order in the algebra
- If is the group ring of a finite group , then is an -order on

A fundamental property of -orders is that every element of an -order is integral over .^{[3]}

If the integral closure of in is an -order then the integrality of every element of every -order shows that must be the unique maximal -order in . However need not always be an -order: indeed need not even be a ring, and even if is a ring (for example, when is commutative) then need not be an -lattice.^{[3]}

## Algebraic number theory[edit]

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 complex numbers of the form , with and integers.*^{[4]}

The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.

## See also[edit]

- Hurwitz quaternion order – An example of ring order

## Notes[edit]

## References[edit]

- Pohst, M.; Zassenhaus, H. (1989).
*Algorithmic Algebraic Number Theory*. Encyclopedia of Mathematics and its Applications. Vol. 30. Cambridge University Press. ISBN 0-521-33060-2. Zbl 0685.12001. - Reiner, I. (2003).
*Maximal Orders*. London Mathematical Society Monographs. New Series. Vol. 28. Oxford University Press. ISBN 0-19-852673-3. Zbl 1024.16008.