Order (ring theory) explained

l{O} of a ring A , such that

A is a finite-dimensional algebra over the field Q of rational numbersl{O} spans A over Q , andl{O} is a Z -lattice in A .

The last two conditions can be stated in less formal terms: Additively,

l{O} is a free abelian group generated by a basis for A over Q .

More generally for

R an integral domain with fraction field K , an R -order in a finite-dimensional K -algebra A is a subring l{O} of A which is a full R -lattice; i.e. is a finite R -module with the property that l{O} ⊗ _{RK=A} .^[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 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:^[2]

• If

A is the matrix ring M _{n(K)} over K , then the matrix ring M _{n(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 the integrality of every element of every R -order shows that S must be the unique maximal R -order in A . However S need not always be an R -order: 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 l{O} 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 complex numbers of the form a +2 bi , with a and b 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

• Hurwitz quaternion order – An example of ring order

References

• Book: Pohst . M. . Zassenhaus . H. . Hans Zassenhaus . Algorithmic Algebraic Number Theory . Encyclopedia of Mathematics and its Applications . 30 . . 1989 . 0-521-33060-2 . 0685.12001 .
• Book: Reiner, I. . Irving Reiner

. Irving Reiner . Maximal Orders . London Mathematical Society Monographs. New Series . 28 . . 2003 . 0-19-852673-3 . 1024.16008 .

Notes and References

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