Rahul Sharma (Editor)

Order (ring theory)

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

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

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

More generally for R an integral domain contained in a field K we define O 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 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.

Examples:

  • If A is the matrix ring Mn(K) over K then the matrix ring Mn(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[a] is an R-order in the algebra K[a]
  • If A is the group ring K[G] of a finite group G then R[G] is an R-order on K[G]

    • 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.

    Algebraic number theory

    The leading example is the case where A is a number field K and O is its ring of integers. In algebraic number theory there are examples for any K other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension A=Q(i) of Gaussian rationals over Q, the integral closure of Z is the ring of Gaussian integers Z[i] and so this is the unique maximal Z-order: all other orders in A are contained in it: for example, we can take the subring of the

    a + b i ,

    for which b is an even number.

    The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.

    References

    Order (ring theory) Wikipedia
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