order (ring theory)

{{Redirect|Maximal order|the maximal order of an arithmetic function|Extremal orders of an arithmetic function}}

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 $\mathcal{O}\otimes_RK=A$ .Reiner (2003) p. 108

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:Reiner (2003) pp. 108–109

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 $R$ .Reiner (2003) p. 110

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 $\mathcal{O}$ 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.Pohst and Zassenhaus (1989) p. 22

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

References

{{cite book | last1=Pohst | first1=M. | last2=Zassenhaus | first2=H. | author2-link=Hans Zassenhaus | title=Algorithmic Algebraic Number Theory | series=Encyclopedia of Mathematics and its Applications | volume=30 | publisher=Cambridge University Press | year=1989 | isbn=0-521-33060-2 | zbl=0685.12001 }}
{{cite book | last=Reiner | first=I. | authorlink=Irving Reiner | title=Maximal Orders | series=London Mathematical Society Monographs. New Series | volume=28 | publisher=Oxford University Press | year=2003 | isbn=0-19-852673-3 | zbl=1024.16008 }}

Category:Ring theory

order (ring theory)

Examples

Algebraic number theory

See also

Notes

References