# Order (ring theory)

In mathematics, an **order** in the sense of ring theory is a subring of a ring , such that

*is a finite-dimensional algebra over the field of rational numbers*- spans
*over , and* - is a -lattice in
*.*

The last two conditions can be stated in less formal terms: Additively, is a free abelian group generated by a basis for * over .*

More generally for * an integral domain contained in a field **, we define to be an **-order in a **-algebra ** if it is a subring of ** which is a full **-lattice.[1]*

When * 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.