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Concept
Order (ring theory)
Summary
In mathematics, an order in the sense of ring theory is a subring of a ring , such that
is a finite-dimensional algebra over the field of rational numbers
spans over , and
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.
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.
Some examples of orders are:
If is the matrix ring over , then the matrix ring over is an -order in
If is an integral domain and a finite separable extension of , then the integral closure of in is an -order in .
If in is an integral element over , then the polynomial ring is an -order in the algebra
If is the group ring of a finite group , then is an -order on
A fundamental property of -orders is that every element of an -order is integral over .
If the integral closure of in is an -order then this result shows that must be the maximal -order in . However this hypothesis is not always satisfied: indeed need not even be a ring, and even if is a ring (for example, when is commutative) then need not be an -lattice.
The leading example is the case where is a number field and is its ring of integers. In algebraic number theory there are examples for any other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension of Gaussian rationals over , the integral closure of is the ring of Gaussian integers and so this is the unique maximal -order: all other orders in are contained in it.
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