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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 with fraction field $K$ , an $R$ -order in a finite-dimensional $K$ -algebra $A$ is a subring ${\mathcal {O}}$ of $A$ which is a full $R$ -lattice; i.e. is a finite $R$ -module with the property that ${\mathcal {O}}\otimes _{R}K=A$ .^[1]

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:^[2]

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 $R$ .^[3]

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.^[3]

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.^[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.

Notes

^ Reiner (2003) p. 108
^ Reiner (2003) pp. 108–109
^ ^a ^b Reiner (2003) p. 110
^ Pohst and Zassenhaus (1989) p. 22

References

Pohst, M.; Zassenhaus, H. (1989). Algorithmic Algebraic Number Theory. Encyclopedia of Mathematics and its Applications. Vol. 30. Cambridge University Press. ISBN 0-521-33060-2. Zbl 0685.12001.
Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series. Vol. 28. Oxford University Press. ISBN 0-19-852673-3. Zbl 1024.16008.

[1] Reiner (2003) p. 108

[2] Reiner (2003) pp. 108–109

[R110-3] Reiner (2003) p. 110

[4] Pohst and Zassenhaus (1989) p. 22

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[2]

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