Order (ring theory)

In mathematics, an order in the sense of ring theory is a subring [math]\displaystyle{ \mathcal{O} }[/math] of a ring [math]\displaystyle{ A }[/math], such that

1. [math]\displaystyle{ A }[/math] is a finite-dimensional algebra over the field [math]\displaystyle{ \mathbb{Q} }[/math] of rational numbers
2. [math]\displaystyle{ \mathcal{O} }[/math] spans [math]\displaystyle{ A }[/math] over [math]\displaystyle{ \mathbb{Q} }[/math], and
3. [math]\displaystyle{ \mathcal{O} }[/math] is a [math]\displaystyle{ \mathbb{Z} }[/math]-lattice in [math]\displaystyle{ A }[/math].

The last two conditions can be stated in less formal terms: Additively, [math]\displaystyle{ \mathcal{O} }[/math] is a free abelian group generated by a basis for [math]\displaystyle{ A }[/math] over [math]\displaystyle{ \mathbb{Q} }[/math].

More generally for [math]\displaystyle{ R }[/math] an integral domain contained in a field [math]\displaystyle{ K }[/math], we define [math]\displaystyle{ \mathcal{O} }[/math] to be an [math]\displaystyle{ R }[/math]-order in a [math]\displaystyle{ K }[/math]-algebra [math]\displaystyle{ A }[/math] if it is a subring of [math]\displaystyle{ A }[/math] which is a full [math]\displaystyle{ R }[/math]-lattice.^[1]

When [math]\displaystyle{ 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:^[2]

• If [math]\displaystyle{ A }[/math] is the matrix ring [math]\displaystyle{ M_n(K) }[/math] over [math]\displaystyle{ K }[/math], then the matrix ring [math]\displaystyle{ M_n(R) }[/math] over [math]\displaystyle{ R }[/math] is an [math]\displaystyle{ R }[/math]-order in [math]\displaystyle{ A }[/math]
• If [math]\displaystyle{ R }[/math] is an integral domain and [math]\displaystyle{ L }[/math] a finite separable extension of [math]\displaystyle{ K }[/math], then the integral closure [math]\displaystyle{ S }[/math] of [math]\displaystyle{ R }[/math] in [math]\displaystyle{ L }[/math] is an [math]\displaystyle{ R }[/math]-order in [math]\displaystyle{ L }[/math].
• If [math]\displaystyle{ a }[/math] in [math]\displaystyle{ A }[/math] is an integral element over [math]\displaystyle{ R }[/math], then the polynomial ring [math]\displaystyle{ R[a] }[/math] is an [math]\displaystyle{ R }[/math]-order in the algebra [math]\displaystyle{ K[a] }[/math]
• If [math]\displaystyle{ A }[/math] is the group ring [math]\displaystyle{ K[G] }[/math] of a finite group [math]\displaystyle{ G }[/math], then [math]\displaystyle{ R[G] }[/math] is an [math]\displaystyle{ R }[/math]-order on [math]\displaystyle{ K[G] }[/math]

A fundamental property of [math]\displaystyle{ R }[/math]-orders is that every element of an [math]\displaystyle{ R }[/math]-order is integral over [math]\displaystyle{ R }[/math].^[3]

If the integral closure [math]\displaystyle{ S }[/math] of [math]\displaystyle{ R }[/math] in [math]\displaystyle{ A }[/math] is an [math]\displaystyle{ R }[/math]-order then this result shows that [math]\displaystyle{ S }[/math] must be the^{[clarification needed]} maximal [math]\displaystyle{ R }[/math]-order in [math]\displaystyle{ A }[/math]. However this hypothesis is not always satisfied: indeed [math]\displaystyle{ S }[/math] need not even be a ring, and even if [math]\displaystyle{ S }[/math] is a ring (for example, when [math]\displaystyle{ A }[/math] is commutative) then [math]\displaystyle{ S }[/math] need not be an [math]\displaystyle{ R }[/math]-lattice.^[3]

Algebraic number theory

The leading example is the case where [math]\displaystyle{ A }[/math] is a number field [math]\displaystyle{ K }[/math] and [math]\displaystyle{ \mathcal{O} }[/math] is its ring of integers. In algebraic number theory there are examples for any [math]\displaystyle{ 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]\displaystyle{ A=\mathbb{Q}(i) }[/math] of Gaussian rationals over [math]\displaystyle{ \mathbb{Q} }[/math], the integral closure of [math]\displaystyle{ \mathbb{Z} }[/math] is the ring of Gaussian integers [math]\displaystyle{ \mathbb{Z}[i] }[/math] and so this is the unique maximal [math]\displaystyle{ \mathbb{Z} }[/math]-order: all other orders in [math]\displaystyle{ A }[/math] are contained in it. For example, we can take the subring of complex numbers of the form [math]\displaystyle{ a+2bi }[/math], with [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] 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

Notes

References

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

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