Order (ring Theory)

Order (ring Theory)

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

1. A is a ring which is a finite-dimensional algebra over the rational number field
2. spans A over, so that, and
3. is a Z-lattice in A.

The last two conditions condition can be stated in less formal terms, that additively should be a free abelian group generated by a basis for A over .

More generally for R an integral domain contained in a field K we define to be an R-order in a K-algebra A if it is a subring of A which is a full R-lattice.

When A is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions are 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 maximum orders: 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:

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

A fundamental property of R-orders is that every element of an R-order is integral over R.

If the integral closure S of R in A is an R-order then this result shows that S must be the maximal R-order in A. However this is not always the case: indeed S need not even be a ring, and even if S is a ring (for example, when A is commutative) then S need not be an R-lattice.