Order (ring theory)

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

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.

Examples

Some examples of orders are:

• If ${\displaystyle A}$ is the matrix ring ${\displaystyle M_n(K)}$ over ${\displaystyle K}$, then the matrix ring ${\displaystyle M_n(R)}$ over ${\displaystyle R}$ is an ${\displaystyle R}$-order in ${\displaystyle A}$
• If ${\displaystyle R}$ is an integral domain and ${\displaystyle L}$ a finite separable extension of ${\displaystyle K}$, then the integral closure ${\displaystyle S}$ of ${\displaystyle R}$ in ${\displaystyle L}$ is an ${\displaystyle R}$-order in ${\displaystyle L}$.
• If ${\displaystyle a}$ in ${\displaystyle A}$ is an integral element over ${\displaystyle R}$, then the polynomial ring ${\displaystyle R[a]}$ is an ${\displaystyle R}$-order in the algebra ${\displaystyle K[a]}$
• If ${\displaystyle A}$ is the group ring ${\displaystyle K[G]}$ of a finite group ${\displaystyle G}$, then ${\displaystyle R[G]}$ is an ${\displaystyle R}$-order on ${\displaystyle K[G]}$

A fundamental property of ${\displaystyle R}$-orders is that every element of an ${\displaystyle R}$-order is integral over ${\displaystyle R}$.

If the integral closure ${\displaystyle S}$ of ${\displaystyle R}$ in ${\displaystyle A}$ is an ${\displaystyle R}$-order then the integrality of every element of every ${\displaystyle R}$-order shows that ${\displaystyle S}$ must be the unique maximal ${\displaystyle R}$-order in ${\displaystyle A}$. However ${\displaystyle S}$ need not always be an ${\displaystyle R}$-order: indeed ${\displaystyle S}$ need not even be a ring, and even if ${\displaystyle S}$ is a ring (for example, when ${\displaystyle A}$ is commutative) then ${\displaystyle S}$ need not be an ${\displaystyle R}$-lattice.