Order (ring theory)

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

1. A is a ring which is a finite-dimensional algebra over the rational number field $\mathbb {Q}$
2. ${\mathcal {O}}$ spans A over $\mathbb {Q}$, so that $\mathbb {Q} {\mathcal {O}}=A$, and
3. ${\mathcal {O}}$ is a Z-lattice in A.

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

More generally for R an integral domain contained in a field K we define ${\mathcal {O}}$ to be an R-order in a K-algebra A if it is a subring of A which is a full R-lattice.^[1]

When 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.

Examples:^[2]

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

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

If the integral closure S of R in A is an R-order then this result shows that S must be the maximal R-order in A. However this is not always the case: indeed S need not even be a ring, and even if S is a ring (for example, when A is commutative) then S need not be an R-lattice.^[3]

Algebraic number theory

The leading example is the case where A 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 field extension A=Q(i) of Gaussian rationals over Q, the integral closure of Z is the ring of Gaussian integers Z[i] and so this is the unique maximal Z-order: all other orders in A are contained in it: for example, we can take the subring of the

$a+bi,$

for which b is an even number.^[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.

See also

• Hurwitz quaternion order - An example of ring order

References

1. Reiner (2003) p.108
2. Reiner (2003) pp.108–109
3. Reiner (2003) p.110
4. Pohst&Zassenhaus (1989) p.22

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