Order (ring theory)

In mathematics, an order in the sense of ring theory is a subring ${\displaystyle {𝒪}}$ of a ring ${\displaystyle A}$, such that

1. ${\displaystyle A}$ is a finite-dimensional algebra over the field ${\displaystyle \mathbb{\mathbb{Q}}}$ of rational numbers
2. ${\displaystyle {𝒪}}$ spans ${\displaystyle A}$ over ${\displaystyle \mathbb{\mathbb{Q}}}$, and
3. ${\displaystyle {𝒪}}$ is a ${\displaystyle \mathbb{\mathbb{Z}}}$-lattice in ${\displaystyle A}$.

The last two conditions can be stated in less formal terms: Additively, ${\displaystyle {𝒪}}$ is a free abelian group generated by a basis for ${\displaystyle A}$ over ${\displaystyle \mathbb{\mathbb{Q}}}$.

More generally for ${\displaystyle R}$ an integral domain contained in a field ${\displaystyle K}$, we define ${\displaystyle {𝒪}}$ to be an ${\displaystyle R}$-order in a ${\displaystyle K}$-algebra ${\displaystyle A}$ if it is a subring of ${\displaystyle A}$ which is a full ${\displaystyle R}$-lattice.^[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 this result shows that ${\displaystyle S}$ must be the^{[clarification needed]} maximal ${\displaystyle R}$-order in ${\displaystyle A}$. However this hypothesis is not always satisfied: 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 ${\displaystyle {𝒪}}$ 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 ${\displaystyle A=\mathbb{\mathbb{Q}}(i)}$ of Gaussian rationals over ${\displaystyle \mathbb{\mathbb{Q}}}$, the integral closure of ${\displaystyle \mathbb{\mathbb{Z}}}$ is the ring of Gaussian integers ${\displaystyle \mathbb{\mathbb{Z}}[i]}$ and so this is the unique maximal ${\displaystyle \mathbb{\mathbb{Z}}}$-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. 30. Cambridge University Press. ISBN 0-521-33060-2. 
• Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series. 28. Oxford University Press. ISBN 0-19-852673-3. 

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