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 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 ${\displaystyle {\mathcal {O))\otimes _{R}K=A}$.^[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.

Examples

Some examples of orders are:^[2]

• 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 ${\displaystyle R}$.^[3]

If the integral closure ${\displaystyle S}$ of ${\displaystyle R}$ in ${\displaystyle A}$ is an ${\displaystyle R}$-order then this result shows that ${\displaystyle S}$ must be the^{[clarification needed]} maximal ${\displaystyle R}$-order in ${\displaystyle A}$. However this hypothesis is not always satisfied: 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.^[3]

Algebraic number theory

The leading example is the case where ${\displaystyle A}$ is a number field ${\displaystyle K}$ and ${\displaystyle {\mathcal {O))}$ is its ring of integers. In algebraic number theory there are examples for any ${\displaystyle K}$ other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension ${\displaystyle A=\mathbb {Q} (i)}$ of Gaussian rationals over ${\displaystyle \mathbb {Q} }$, the integral closure of ${\displaystyle \mathbb {Z} }$ is the ring of Gaussian integers ${\displaystyle \mathbb {Z} [i]}$ and so this is the unique maximal ${\displaystyle \mathbb {Z} }$-order: all other orders in ${\displaystyle A}$ are contained in it. For example, we can take the subring of complex numbers of the form ${\displaystyle a+2bi}$, with ${\displaystyle a}$ and ${\displaystyle b}$ integers.^[4]

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