Order (ring theory)

In mathematics, an order in the sense of ring theory is a subring formula_1 of a ring formula_2, such that
The last two conditions can be stated in less formal terms: Additively, formula_1 is a free abelian group generated by a basis for "formula_2" over formula_4.
More generally for "formula_14" an integral domain contained in a field "formula_15", we define formula_1 to be an "formula_14"-order in a "formula_15"-algebra "formula_2" if it is a subring of "formula_2" which is a full "formula_14"-lattice.
When "formula_2" 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:
A fundamental property of "formula_14"-orders is that every element of an "formula_14"-order is integral over "formula_14".
If the integral closure "formula_33" of "formula_14" in "formula_2" is an "formula_14"-order then this result shows that "formula_33" must be the maximal "formula_14"-order in "formula_2". However this hypothesis is not always satisfied: indeed "formula_33" need not even be a ring, and even if "formula_33" is a ring (for example, when "formula_2" is commutative) then "formula_33" need not be an "formula_14"-lattice.
Algebraic number theory.
The leading example is the case where "formula_2" is a number field "formula_15" and formula_1 is its ring of integers. In algebraic number theory there are examples for any "formula_15" other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension "formula_69" of Gaussian rationals over formula_4, the integral closure of "formula_9" is the ring of Gaussian integers "formula_72" and so this is the unique "maximal" "formula_9"-order: all other orders in "formula_2" are contained in it. For example, we can take the subring of the complex numbers in the form formula_75, with formula_38 and formula_77 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.