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 contained in a field ''$K$'', we define $\mathcal$ 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 /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 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 $\mathcal$ 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