Order (ring theory)

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


 * 1) $$A$$ is a finite-dimensional algebra over the field $$\mathbb{Q}$$ of rational numbers
 * 2) $$\mathcal{O}$$ spans $$A$$ over $$\mathbb{Q}$$, and
 * 3) $$\mathcal{O}$$ is a $$\mathbb{Z}$$-lattice in $$A$$.

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

More generally for $$R$$ an integral domain with fraction field $$K$$, an $$R$$-order in a finite-dimensional $$K$$-algebra $$A$$ is a subring $$\mathcal{O}$$ of $$A$$ which is a full $$R$$-lattice; i.e. is a finite $$R$$-module with the property that $$\mathcal{O}\otimes_RK=A$$.

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
Some examples of orders are:
 * If $$A$$ is the matrix ring $$M_n(K)$$ over $$K$$, then the matrix ring $$M_n(R)$$ 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 $$R$$.

If the integral closure $$S$$ of $$R$$ in $$A$$ is an $$R$$-order then the integrality of every element of every $$R$$-order shows that $$S$$ must be the unique maximal $$R$$-order in $$A$$. However $$S$$ need not always be an $$R$$-order: 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.

Algebraic number theory
The leading example is the case where $$A$$ is a number field $$K$$ and $$\mathcal{O}$$ 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 $$A=\mathbb{Q}(i)$$ of Gaussian rationals over $$\mathbb{Q}$$, the integral closure of $$\mathbb{Z}$$ is the ring of Gaussian integers $$\mathbb{Z}[i]$$ and so this is the unique maximal $$\mathbb{Z}$$-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.

The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.