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

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 ${\displaystyle A}$ over ${\displaystyle \mathbb {Q} }$.

More generally for ${\displaystyle R}$ an integral domain contained in a field ${\displaystyle K}$, we define ${\displaystyle {\mathcal {O}}}$ to be an ${\displaystyle R}$-order in a ${\displaystyle K}$-algebra ${\displaystyle A}$ if it is a subring of ${\displaystyle A}$ which is a full ${\displaystyle R}$-lattice.[1]

When ${\displaystyle A}$ 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.