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Order (ring theory)

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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.


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[edit]

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[edit]


  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


  • 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.