Order (ring theory)

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"Maximal order" redirects here. For the maximal order of an arithmetic function, see Extremal orders of an arithmetic function.

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

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

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

More generally for R an integral domain contained in a field K we define \mathcal{O} to be an R-order in a K-algebra A if it is a subring of A which is a full R-lattice.[1]

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:[2]

A fundamental property of R-orders is that every element of an R-order is integral over R.[3]

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.[3]

Algebraic number theory[edit]

The leading example is the case where A is a number field K and \mathcal{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+bi,

for which b is an even number.[4]

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

See also[edit]

References[edit]

  1. ^ Reiner (2003) p.108
  2. ^ Reiner (2003) pp.108–109
  3. ^ a b Reiner (2003) p.110
  4. ^ Pohst&Zassenhaus (1989) p.22