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 $1$).

See also

• Hurwitz quaternion order - An example of ring order