In mathematics, an order in the sense of ring theory is a subring of a ring , such that

  1. is a finite-dimensional algebra over the field of rational numbers
  2. spans over , and
  3. is a -lattice in .

The last two conditions can be stated in less formal terms: Additively, is a free abelian group generated by a basis for over .

More generally for an integral domain with fraction field , an -order in a finite-dimensional -algebra is a subring of which is a full -lattice; i.e. is a finite -module with the property that .[1]

When 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

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Some examples of orders are:[2]

  • If   is the matrix ring   over  , then the matrix ring   over   is an  -order in  
  • If   is an integral domain and   a finite separable extension of  , then the integral closure   of   in   is an  -order in  .
  • If   in   is an integral element over  , then the polynomial ring   is an  -order in the algebra  
  • If   is the group ring   of a finite group  , then   is an  -order on  

A fundamental property of  -orders is that every element of an  -order is integral over  .[3]

If the integral closure   of   in   is an  -order then the integrality of every element of every  -order shows that   must be the unique maximal  -order in  . However   need not always be an  -order: indeed   need not even be a ring, and even if   is a ring (for example, when   is commutative) then   need not be an  -lattice.[3]

Algebraic number theory

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The leading example is the case where   is a number field   and   is its ring of integers. In algebraic number theory there are examples for any   other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension   of Gaussian rationals over  , the integral closure of   is the ring of Gaussian integers   and so this is the unique maximal  -order: all other orders in   are contained in it. For example, we can take the subring of complex numbers of the form  , with   and   integers.[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

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Notes

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  1. ^ Reiner (2003) p. 108
  2. ^ Reiner (2003) pp. 108–109
  3. ^ a b Reiner (2003) p. 110
  4. ^ Pohst and Zassenhaus (1989) p. 22

References

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  • Pohst, M.; Zassenhaus, H. (1989). Algorithmic Algebraic Number Theory. Encyclopedia of Mathematics and its Applications. Vol. 30. Cambridge University Press. ISBN 0-521-33060-2. Zbl 0685.12001.
  • Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series. Vol. 28. Oxford University Press. ISBN 0-19-852673-3. Zbl 1024.16008.