Order (ring theory)

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

1. A is a finite-dimensional algebra over the field ℚ of rational numbers
2. 𝒪 spans A over ℚ, and
3. 𝒪 is a ℤ-lattice in A.

The last two conditions can be stated in less formal terms: Additively, 𝒪 is a free abelian group generated by a basis for A over ℚ.

More generally for R an integral domain contained in a field K, we define 𝒪 to be an R-order in a K-algebra A if it is a subring of A which is a full R-lattice.[1]

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.

## Contents

* 1 Examples
* 2 Algebraic number theory
* 3 See also
* 4 Notes
* 5 References

## Examples

Some examples of orders are:[2]

* If A is the matrix ring Mn(K) over K, then the matrix ring Mn(R) over R is an R-order in A
* If R is an integral domain and L a finite separable extension of K, then the integral closure S of R in L is an R-order in L.
* If a in A is an integral element over R, then the polynomial ring R[a] is an R-order in the algebra K[a]
* If A is the group ring K[G] of a finite group G, then R[G] is an R-order on K[G]

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

If the integral closure S of R in A is an R-order then this result shows that S must be the[clarification needed] maximal R-order in A. However this hypothesis is not always satisfied: 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.[3]

## Algebraic number theory

The leading example is the case where A is a number field K and 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 A=ℚ(i) of Gaussian rationals over ℚ, the integral closure of ℤ is the ring of Gaussian integers ℤ[i] and so this is the unique maximal 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.[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

1. ↑ Reiner (2003) p. 108
2. ↑ Reiner (2003) pp. 108–109
3. ↑ 3.0 3.1 Reiner (2003) p. 110
4. ↑ Pohst and Zassenhaus (1989) p. 22

## 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.
* Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series. 28. Oxford University Press. ISBN 0-19-852673-3.

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