Editing 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]] <math>\mathcal{O}</math> of a [[ring (mathematics)|ring]] <math>A</math>, such that

#''<math>A</math>'' is a finite-dimensional [[Algebra over a field|algebra]] over the [[Field (mathematics)|field]] <math>\mathbb{Q}</math> of [[rational number]]s
#<math>\mathcal{O}</math> spans ''<math>A</math>'' over <math>\mathbb{Q}</math>, and
#<math>\mathcal{O}</math> is a <math>\mathbb{Z}</math>-[[lattice (module)|lattice]] in ''<math>A</math>''.

The last two conditions can be stated in less formal terms:  Additively, <math>\mathcal{O}</math> is a [[free abelian group]] generated by a [[basis (linear algebra)|basis]] for ''<math>A</math>'' over <math>\mathbb{Q}</math>.

More generally for ''<math>R</math>'' an [[integral domain]] with fraction field ''<math>K</math>'', an ''<math>R</math>''-order in a finite-dimensional ''<math>K</math>''-algebra ''<math>A</math>'' is a subring <math>\mathcal{O}</math> of ''<math>A</math>'' which is a full ''<math>R</math>''-lattice; i.e. is a finite ''<math>R</math>''-module with the property that ''<math>\mathcal{O}\otimes_RK=A</math>''.<ref>Reiner (2003) p.&nbsp;108</ref>

When ''<math>A</math>'' is not a [[commutative ring]], the idea of order is still important, but the phenomena are different. For example, the [[Hurwitz quaternion]]s form a '''maximal''' order in the [[quaternion]]s 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 ring]]s.

==Examples==
Some examples of orders are:<ref>Reiner (2003) pp.&nbsp;108–109</ref>
* If <math>A</math> is the [[matrix ring]] <math>M_n(K)</math> over <math>K</math>, then the matrix ring <math>M_n(R)</math> over <math>R</math> is an <math>R</math>-order in <math>A</math>
* If <math>R</math> is an integral domain and <math>L</math> a finite [[separable extension]] of <math>K</math>, then the [[integral closure]] <math>S</math> of <math>R</math> in <math>L</math> is an <math>R</math>-order in <math>L</math>.
* If <math>a</math> in <math>A</math> is an [[integral element]] over <math>R</math>, then the [[polynomial ring]] <math>R[a]</math> is an <math>R</math>-order in the algebra <math>K[a]</math>
* If <math>A</math> is the [[group ring]] <math>K[G]</math> of a [[finite group]] <math>G</math>, then <math>R[G]</math> is an <math>R</math>-order on <math>K[G]</math>

A fundamental property of <math>R</math>-orders is that every element of an <math>R</math>-order is [[integral element|integral]] over <math>R</math>.<ref name=R110>Reiner (2003) p.&nbsp;110</ref>

If the integral closure <math>S</math> of <math>R</math> in <math>A</math> is an <math>R</math>-order then the integrality of every element of every <math>R</math>-order shows that <math>S</math> must be the unique maximal <math>R</math>-order in <math>A</math>.  However <math>S</math> need not always be an <math>R</math>-order: indeed <math>S</math> need not even be a ring, and even if <math>S</math> is a ring (for example, when <math>A</math> is commutative) then <math>S</math> need not be an <math>R</math>-lattice.<ref name=R110/>

==Algebraic number theory==
The leading example is the case where ''<math>A</math>'' is a [[number field]] ''<math>K</math>'' and <math>\mathcal{O}</math> is its [[ring of integers]]. In [[algebraic number theory]] there are examples for any ''<math>K</math>'' other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the [[field extension]] ''<math>A=\mathbb{Q}(i)</math>'' of [[Gaussian rational]]s over <math>\mathbb{Q}</math>, the integral closure of ''<math>\mathbb{Z}</math>''  is the ring of [[Gaussian integer]]s ''<math>\mathbb{Z}[i]</math>'' and so this is the unique ''maximal'' ''<math>\mathbb{Z}</math>''-order: all other orders in ''<math>A</math>'' are contained in it. For example, we can take the subring of [[complex number]]s of the form <math>a+2bi</math>, with <math>a</math> and <math>b</math> integers.<ref>Pohst and Zassenhaus (1989) p.&nbsp;22</ref>

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==
{{reflist}}
==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]]