Order (ring theory)

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 ${\displaystyle {\mathcal {O}}\otimes _{R}K=A}$.^[1]

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:^[2]

• If ${\displaystyle A}$ is the matrix ring ${\displaystyle M_{n}(K)}$ over ${\displaystyle K}$, then the matrix ring ${\displaystyle M_{n}(R)}$ over ${\displaystyle R}$ is an ${\displaystyle R}$-order in ${\displaystyle A}$
• If ${\displaystyle R}$ is an integral domain and ${\displaystyle L}$ a finite separable extension of ${\displaystyle K}$, then the integral closure ${\displaystyle S}$ of ${\displaystyle R}$ in ${\displaystyle L}$ is an ${\displaystyle R}$-order in ${\displaystyle L}$.
• If ${\displaystyle a}$ in ${\displaystyle A}$ is an integral element over ${\displaystyle R}$, then the polynomial ring ${\displaystyle R[a]}$ is an ${\displaystyle R}$-order in the algebra ${\displaystyle K[a]}$
• If ${\displaystyle A}$ is the group ring ${\displaystyle K[G]}$ of a finite group ${\displaystyle G}$, then ${\displaystyle R[G]}$ is an ${\displaystyle R}$-order on ${\displaystyle K[G]}$

A fundamental property of ${\displaystyle R}$-orders is that every element of an ${\displaystyle R}$-order is integral over ${\displaystyle R}$.^[3]

If the integral closure ${\displaystyle S}$ of ${\displaystyle R}$ in ${\displaystyle A}$ is an ${\displaystyle R}$-order then the integrality of every element of every ${\displaystyle R}$-order shows that ${\displaystyle S}$ must be the unique maximal ${\displaystyle R}$-order in ${\displaystyle A}$. However ${\displaystyle S}$ need not always be an ${\displaystyle R}$-order: indeed ${\displaystyle S}$ need not even be a ring, and even if ${\displaystyle S}$ is a ring (for example, when ${\displaystyle A}$ is commutative) then ${\displaystyle S}$ need not be an ${\displaystyle R}$-lattice.^[3]

The leading example is the case where ${\displaystyle A}$ is a number field ${\displaystyle K}$ and ${\displaystyle {\mathcal {O}}}$ is its ring of integers. In algebraic number theory there are examples for any ${\displaystyle K}$ other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension ${\displaystyle A=\mathbb {Q} (i)}$ of Gaussian rationals over ${\displaystyle \mathbb {Q} }$, the integral closure of ${\displaystyle \mathbb {Z} }$ is the ring of Gaussian integers ${\displaystyle \mathbb {Z} [i]}$ and so this is the unique maximal ${\displaystyle \mathbb {Z} }$-order: all other orders in ${\displaystyle A}$ are contained in it. For example, we can take the subring of complex numbers of the form ${\displaystyle a+2bi}$, with ${\displaystyle a}$ and ${\displaystyle b}$ integers.^[4]

The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.

• Hurwitz quaternion order – An example of ring order

• Pohst, M.; Zassenhaus, H. (1989). Algorithmic Algebraic Number Theory. Encyclopedia of Mathematics and its Applications. Vol. 30. Cambridge University Press. ISBN 0-521-33060-2. Zbl 0685.12001.
• Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series. Vol. 28. Oxford University Press. ISBN 0-19-852673-3. Zbl 1024.16008.