Maximal order - Knowlpedia

An $R$ -order $\mathcal O$ in an algebra $A$ is maximal if every $R$ -order $\mathcal O'$ with $\mathcal O\subseteq\mathcal O'\subseteq A$ equals $\mathcal O$ .

The base ring and ambient algebra are fixed. Maximality is with respect to inclusion among orders, not among all subrings of $A$ .

Number fields

For a number field $K$ , its ring of integers $\mathcal O_K$ is the unique maximal $\mathbb Z$ -order: every order consists of algebraic integers, and $\mathcal O_K$ is itself a full lattice.

An order in $K$ is a Dedekind domain exactly when it equals $\mathcal O_K$ . Every such order is Noetherian of dimension one; the missing condition for a nonmaximal order is integral closedness.

Noncommutative algebras

Maximal orders need not be unique as embedded subrings. For example, $M_2(\mathbb Z)$ and $gM_2(\mathbb Z)g^{-1}$ , with $g=\operatorname{diag}(2,1)$ , are distinct maximal orders in $M_2(\mathbb Q)$ .

To see maximality of $M_2(\mathbb Z)$ , an overorder containing a matrix $x$ also contains $E_{ai}xE_{ja}=x_{ij}E_{aa}$ . Each rational coefficient $x_{ij}$ must be integral over $\mathbb Z$ , hence an integer. Conjugation preserves maximality.

The number of conjugacy classes of maximal orders is a type number in the quaternion setting.

References

John Voight, Quaternion Algebras, author's text, Definition 10.4.1 and §10.5.