Combinatorial number system: Difference between revisions
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The | {{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 | |||
#''A'' is a ring which is a finite-dimensional [[Algebra over a field|algebra]] over the [[rational number field]] <math>\mathbb{Q}</math> | |||
#<math>\mathcal{O}</math> spans ''A'' over <math>\mathbb{Q}</math>, so that <math>\mathbb{Q} \mathcal{O} = A</math>, and | |||
#<math>\mathcal{O}</math> is a '''Z'''-[[lattice (module)|lattice]] in ''A''. | |||
The last two conditions condition can be stated in less formal terms: Additively, <math>\mathcal{O}</math> is a [[free abelian group]] generated by a basis for ''A'' over <math>\mathbb{Q}</math>. | |||
More generally for ''R'' an integral domain contained in a field ''K'' we define <math>\mathcal{O}</math> to be an ''R''-order in a ''K''-algebra ''A'' if it is a subring of ''A'' which is a full ''R''-lattice.<ref>Reiner (2003) p.108</ref> | |||
When ''A'' 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 '''maximum orders''': 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:<ref>Reiner (2003) pp.108–109</ref> | |||
* If ''A'' is the [[matrix ring]] ''M''<sub>''n''</sub>(''K'') over ''K'' then the matrix ring ''M''<sub>''n''</sub>(''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''[''a''] is an ''R''-order in the algebra ''K''[''a''] | |||
* If ''A'' is the [[group ring]] ''K''[''G''] of a finite group ''G'' then ''R''[''G''] is an ''R''-order on ''K''[''G''] | |||
A fundamental property of ''R''-orders is that every element of an ''R''-order is integral over ''R''.<ref name=R110>Reiner (2003) p.110</ref> | |||
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 is not always the case: 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.<ref name=R110/> | |||
==Algebraic number theory== | |||
The leading example is the case where ''A'' is a [[number field]] ''K'' and <math>\mathcal{O}</math> 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''='''Q'''(i) of [[Gaussian rational]]s over '''Q''', the integral closure of '''Z''' is the ring of [[Gaussian integer]]s '''Z'''[i] and so this is the unique ''maximal'' '''Z'''-order: all other orders in ''A'' are contained in it: for example, we can take the subring of the | |||
:<math>a+bi,</math> | |||
for which ''b'' is an [[even number]].<ref>Pohst&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 | |||
==References== | |||
{{reflist}} | |||
* {{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]] | |||
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In mathematics, an order in the sense of ring theory is a subring of a ring , such that
- A is a ring which is a finite-dimensional algebra over the rational number field
- spans A over , so that , and
- is a Z-lattice in A.
The last two conditions condition can be stated in less formal terms: Additively, is a free abelian group generated by a basis for A over .
More generally for R an integral domain contained in a field K we define to be an R-order in a K-algebra A if it is a subring of A which is a full R-lattice.[1]
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 maximum orders: 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:[2]
- If A is the matrix ring Mn(K) over K then the matrix ring Mn(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[a] is an R-order in the algebra K[a]
- If A is the group ring K[G] of a finite group G then R[G] is an R-order on K[G]
A fundamental property of R-orders is that every element of an R-order is integral over R.[3]
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 is not always the case: 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.[3]
Algebraic number theory
The leading example is the case where A is a number field K and 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=Q(i) of Gaussian rationals over Q, the integral closure of Z is the ring of Gaussian integers Z[i] and so this is the unique maximal Z-order: all other orders in A are contained in it: for example, we can take the subring of the
for which b is an even number.[4]
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
References
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