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 contained in a field **, we define to be an **-order in a **-algebra ** if it is a subring of ** which is a full **-lattice.*^{[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

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 this result shows that ** must be the*^{[clarification needed]} maximal *-order in **. However this hypothesis is not always satisfied: 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

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.*^{[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

- Hurwitz quaternion order – An example of ring order

## Notes

## References

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