# 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 contained in a field $K$ , we define ${\mathcal {O}}$ to be an $R$ -order in a $K$ -algebra $A$ if it is a subring of $A$ which is a full $R$ -lattice.

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.