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 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:
 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 .

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 (for example, when  is commutative) 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 complex numbers of 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.

See also 
 Hurwitz quaternion order – An example of ring order

Notes

References
 
 

Ring theory