Order (ring theory)

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 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 ${\displaystyle {\mathcal {O}}}$ 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.