Order (ring Theory)

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

#''$A$'' is a finite-dimensional algebra over the field $\mathbb$ of rational numbers
#$\mathcal$ spans ''$A$'' over $\mathbb$, and
#$\mathcal$ is a $\mathbb$- lattice in ''$A$''.

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

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

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 /math> is an $R$-order in the algebra K /math>
* If $A$ is the group ring K /math> of a finite group $G$, then R /math> is an $R$-order on K /math>

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 $\mathcal$ 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 ''$A=\mathbb(i)$'' of Gaussian rationals over $\mathbb$, the integral closure of ''$\mathbb$'' is the ring of Gaussian integers ''\mathbb /math>'' and so this is the unique ''maximal'' ''$\mathbb$''-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.

See also

* Hurwitz quaternion order – An example of ring order

Notes

References

*
* {{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

Ring theory