In mathematics, an order in the sense of ring theory is a subring
, such that The last two conditions can be stated in less formal terms: Additively,
is a free abelian group generated by a basis for
an integral domain with fraction field
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.
Some examples of orders are:[2] A fundamental property of
[3] If the integral closure
-order then the integrality of every element of every
must be the unique maximal
[3] The leading example is the case where
is a number field
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
, the integral closure of
is the ring of Gaussian integers
and so this is the unique maximal
For example, we can take the subring of complex numbers of the form
[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.