Order (ring theory)

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

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

The third condition can be stated more accurately, in terms of the extension of scalars of R to the real numbers, embedding R in a real vector space (equivalently, taking the tensor product over \mathbb{Q}). In less formal terms, additively O should be a free abelian group generated by a basis for R over \mathbb{Q}.

The leading example is the case where R 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 Gaussian integers we can take the subring of the

a + bi,

for which b is an even number. A basic result on orders states that the ring of integers in K is the unique maximal order: all other orders in K are contained in it.

When R 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.

Because there is a local-global principle for lattices the maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.

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