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{{Redirect|Maximal order|the maximal order of an arithmetic function|Extremal orders of an arithmetic function}}
In [[mathematics]], certain subsets of some [[Field (mathematics)|fields]] are called '''orders'''. The set of [[integer]]s is an order in the [[rational numbers]] (the only one). In an [[algebraic number field]] {{tmath|K}}, an order is a [[ring of algebraic integers]] whose [[field of fractions]] is {{tmath|K}}, and the '''maximal order''', often denoted {{tmath|\mathcal O_K}}, is the ring of all algebraic integers in {{tmath|K}}. In a [[local field|non-Archimedean local field]] {{tmath|K}}, an ''order'' is a subring which is generated by finitely many elements of non-negative valuation. In that case, the maximal order, denoted {{tmath|\mathcal O_K}}, is the [[valuation ring]] formed by all elements of non-negative valuation.

Giving the same name to such seemingly different notions is motivated by the [[local–global principle]] that relates properties of a number field with properties of all its local fields.

==Definitions==
The definition of an order is somewhat context-dependent. The simplest definition is in an algebraic number field <math>F</math>, where an order <math>R</math> is a [[subring]] of <math>F</math> that is a finitely-generated <math>\mathbb Z</math>-[[module (mathematics)|module]], which contains a rational [[basis (mathematics)|basis]] of <math>F</math>, i.e., such that <math>\mathbb QR = F.</math>

On the other hand, if <math>F</math> is a non-archimedean [[local field]], an order is a compact-open subring <math>R</math> of <math>F</math>. The maximal order in this case is the [[valuation ring]] of the field.

More generally, which includes both of these special cases, if <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. 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. 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. 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. 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]]

Order (ring theory) - Revision history

imported>Tito Omburo at 22:43, 19 September 2025