Editing Order (ring theory) (section)

{{Redirect|Maximal order|the maximal order of an arithmetic function|Extremal orders of an arithmetic function}}

In [[mathematics]], an '''order''' in the sense of [[ring theory]] is a [[subring]] <math>\mathcal{O}</math> of a [[ring (mathematics)|ring]] <math>A</math>, such that

#''<math>A</math>'' is a finite-dimensional [[Algebra over a field|algebra]] over the [[Field (mathematics)|field]] <math>\mathbb{Q}</math> of [[rational number]]s
#<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 (module)|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 (linear algebra)|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.&nbsp;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 quaternion]]s form a '''maximal''' order in the [[quaternion]]s 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 ring]]s.