Order (ring theory)
# Order (ring theory)

In mathematics, an order in the sense of ring theory is a subring ${\mathcal {O}}$ of a ring $A$, such that

* $A$ is a finite-dimensional algebra over the field $\mathbb {Q}$ of rational numbers
* ${\mathcal {O}}$ spans $A$ over $\mathbb {Q}$, and
* ${\mathcal {O}}$ 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.

## Examples

Some examples of orders are:

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

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

If the integral closure $S$ of $R$ in $A$ is an $R$-order then this result shows that $S$ must be the maximal $R$-order in $A$. However this hypothesis is not always satisfied: indeed $S$ need not even be a ring, and even if $S$ is a ring (for example, when $A$ is commutative) then $S$ need not be an $R$-lattice.

## Algebraic number theory

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

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

