Order (ring theory)

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 $\mathcal{O}$ 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.

Definition

An order R in a number field K is a subring of K which as a $\mathbb{Z}$-module is finitely generated and of maximal rank n = deg(K) (note that we use the "modern" definition of a ring, which includes the existence of the multiplicative identity 1).^[1]

References

1. ^ Henri Cohen, A Course in Computational Algebraic Number Theory 3rd corrected printing, pp. 181, 1996, Springer

See also

• Hurwitz quaternion order - An example of ring order