In mathematics, certain subsets of some fields are called orders. The set of integers is an order in the rational numbers (the only one). In an algebraic number field , an order is a ring of algebraic integers whose field of fractions is , and the maximal order, often denoted , is the ring of all algebraic integers in . In a non-Archimedean local field , an order is a subring which is generated by finitely many elements of non-negative valuation. In that case, the maximal order, denoted , is the valuation ring formed by all elements of non-negative valuation.
Giving the same name to such seemingly different notions is motivated by the local–global principle that relates properties of a number field with properties of all its local fields.
Definitions
The definition of an order is somewhat context-dependent. The simplest definition is in an algebraic number field <math>F</math>, where an order <math>R</math> is a subring of <math>F</math> that is a finitely-generated <math>\mathbb Z</math>-module, which contains a rational basis of <math>F</math>, i.e., such that <math>\mathbb QR = F.</math>
On the other hand, if <math>F</math> is a non-archimedean local field, an order is a compact-open subring <math>R</math> of <math>F</math>. The maximal order in this case is the valuation ring of the field.
More generally, which includes both of these special cases, if <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>.
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.
Examples
Some examples of orders are:
- 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>.
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.
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