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In mathematics, an order in the sense of ring theory is a subring O {\displaystyle {\mathcal {O}}} of a ring A {\displaystyle A} , such that
The last two conditions can be stated in less formal terms: Additively, O {\displaystyle {\mathcal {O}}} is a free abelian group generated by a basis for A {\displaystyle A} over Q {\displaystyle \mathbb {Q} }.
More generally for R {\displaystyle R} an integral domain contained in a field K {\displaystyle K} , we define O {\displaystyle {\mathcal {O}}} to be an R {\displaystyle R} -order in a K {\displaystyle K} -algebra A {\displaystyle A} if it is a subring of A {\displaystyle A} which is a full R {\displaystyle R} -lattice.
When A {\displaystyle 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.