In mathematics, certain subsets of some fields are called orders. The set of integers is an order in the rational numbers (the only one). In an algebraic number field ⁠${\displaystyle K}$⁠, an order is a ring of algebraic integers whose field of fractions is ⁠${\displaystyle K}$⁠, and the maximal order, often denoted ⁠${\displaystyle {\mathcal {O}}_{K}}$⁠, is the ring of all algebraic integers in ⁠${\displaystyle K}$⁠. In a non-Archimedean local field ⁠${\displaystyle K}$⁠, an order is a subring which is generated by finitely many elements of non-negative valuation. In that case, the maximal order, denoted ⁠${\displaystyle {\mathcal {O}}_{K}}$⁠, is the valuation ring formed by all elements of non-negative valuation.

Giving the same name to such seemingly different notions is motivated by the local–global principle that relates properties of a number field with properties of all its local fields.

The definition of an order is somewhat context-dependent. The simplest definition is in an algebraic number field ${\displaystyle F}$, where an order ${\displaystyle R}$ is a subring of ${\displaystyle F}$ that is a finitely-generated ${\displaystyle \mathbb {Z} }$-module, which contains a rational basis of ${\displaystyle F}$, i.e., such that ${\displaystyle \mathbb {Q} R=F.}$

On the other hand, if ${\displaystyle F}$ is a non-archimedean local field, an order is a compact-open subring ${\displaystyle R}$ of ${\displaystyle F}$. The maximal order in this case is the valuation ring of the field.

More generally, which includes both of these special cases, if ${\displaystyle R}$ an integral domain with fraction field ${\displaystyle K}$, an ${\displaystyle R}$-order in a finite-dimensional ${\displaystyle K}$-algebra ${\displaystyle A}$ is a subring ${\displaystyle {\mathcal {O}}}$ of ${\displaystyle A}$ which is a full ${\displaystyle R}$-lattice; i.e. is a finite ${\displaystyle R}$-module with the property that ${\displaystyle {\mathcal {O}}\otimes _{R}K=A}$.

When ${\displaystyle A}$ 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.

Some examples of orders are:

A fundamental property of ${\displaystyle R}$-orders is that every element of an ${\displaystyle R}$-order is integral over ${\displaystyle R}$.