Order (ring theory)
From Wikipedia, the free encyclopedia
In mathematics, an order in the sense of ring theory is a subring O of a ring R that satisfies the conditions
- R is a ring which is a finite-dimensional algebra over the rational number field Q
- O spans R over Q, so that QO = R; and
- 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 Q). In less formal terms, additively O should be a free abelian group generated by a basis for R over Q.
The leading example is the case where R 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 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.
