In mathematics, an order in the sense of ring theory is a subring <math>\mathcal{O}</math> of a ring <math>A</math>, such that
- <math>A</math> is a finite-dimensional algebra over the field <math>\mathbb{Q}</math> of rational numbers
- <math>\mathcal{O}</math> spans <math>A</math> over <math>\mathbb{Q}</math>, and
- <math>\mathcal{O}</math> is a <math>\mathbb{Z}</math>-lattice in <math>A</math>.
The last two conditions can be stated in less formal terms: Additively, <math>\mathcal{O}</math> is a free abelian group generated by a basis for <math>A</math> over <math>\mathbb{Q}</math>.
More generally for <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.
ExamplesEdit
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 theoryEdit
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 alsoEdit
- Hurwitz quaternion order – An example of ring order