Order (ring theory)

In mathematics, an order in the sense of ring theory is a subring of a ring , such that

  1. is a finite-dimensional algebra over the field of rational numbers
  2. spans over , and
  3. is a -lattice in .

The last two conditions can be stated in less formal terms: Additively, is a free abelian group generated by a basis for over .

More generally for an integral domain contained in a field , we define to be an -order in a -algebra if it is a subring of which is a full -lattice.[1]

When 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.

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