# Order (ring theory)

In mathematics, an order in the sense of ring theory is a subring ${\displaystyle {\mathcal {O}}}$ of a ring ${\displaystyle A}$, such that

1. A is a ring which is a finite-dimensional algebra over the rational number field ${\displaystyle \mathbb {Q} }$
2. ${\displaystyle {\mathcal {O}}}$ spans A over ${\displaystyle \mathbb {Q} }$, so that ${\displaystyle \mathbb {Q} {\mathcal {O}}=A}$, and
3. ${\displaystyle {\mathcal {O}}}$ is a Z-lattice in A.

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

More generally for R an integral domain contained in a field K we define ${\displaystyle {\mathcal {O}}}$ 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.

Examples:[2]

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 maximal R-order in A. However this is not always the case: 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 ${\displaystyle {\mathcal {O}}}$ 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 the

${\displaystyle a+bi,}$

for which b is an even number.[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.