{{Redirect|Maximal order|the maximal order of an arithmetic function|Extremal orders of an arithmetic function}} In mathematics, certain subsets of some fields are called '''orders'''. The set of integers 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 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, which contains a rational 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 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:<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 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 rationals over <math>\mathbb{Q}</math>, the integral closure of ''<math>\mathbb{Z}</math>'' is the ring of Gaussian integers ''<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 numbers 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