Order (ring theory)

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

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

The last two conditions can be stated in less formal terms: Additively, ${𝒪}$ is a free abelian group generated by a basis for $A$ over $\mathbb{\mathbb{Q}}$.

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

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 the integrality of every element of every $R$-order shows that $S$ must be the unique maximal $R$-order in $A$. However $S$ need not always be an $R$-order: 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 ${𝒪}$ 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 $A=\mathbb{\mathbb{Q}}(i)$ of Gaussian rationals over $\mathbb{\mathbb{Q}}$, the integral closure of $\mathbb{\mathbb{Z}}$ is the ring of Gaussian integers $\mathbb{\mathbb{Z}}[i]$ and so this is the unique maximal $\mathbb{\mathbb{Z}}$-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.

See also

• Hurwitz quaternion order – An example of ring order