Order in an algebra - Knowlpedia

Let $R$ be a Noetherian integral domain with fraction field $K$ , and let $A\ne0$ be a finite-dimensional associative unital $K$ -algebra. An $R$ -order in $A$ is a subring $\mathcal O\subseteq A$ containing $1_A$ that is a full $R$ -lattice: $\mathcal O$ is a finitely generated $R$ -submodule and $K\mathcal O=A$ .

The scalar action is inherited from $A$ ; in particular $R1_A\subseteq\mathcal O$ . The algebra and its order may be noncommutative.

Z-orders

A $\mathbb Z$ -order is this construction with $R=\mathbb Z$ and $K=\mathbb Q$ . Its additive group is free of rank $\dim_{\mathbb Q}A$ .

Examples are $M_n(\mathbb Z)\subset M_n(\mathbb Q)$ and, for a finite group $G$ , the group ring $\mathbb Z[G]\subset\mathbb Q[G]$ . The matrix units and group elements respectively give the required bases.

Number fields and quaternion algebras

In a number field $K$ , every $\mathbb Z$ -order lies with finite additive index in the ring of integers $\mathcal O_K$ . Indeed, multiplication by any element of the order has an integral matrix on its lattice, so its characteristic polynomial proves integrality.

In $\mathbb Q(i)$ , the rings $\mathbb Z+f\mathbb Zi$ , $f\ge1$ , are orders of index $f$ in $\mathbb Z[i]$ . Noncommutative examples include quaternion orders.

Conventions

Finite generation does not imply freeness over a general Dedekind domain. We use associative multiplication here; a bracket-closed integral form of a Lie algebra is a Lie order.

References

John Voight, Quaternion Algebras, author's text, Definition 10.2.1 and Examples 10.2.3–10.2.4.
Andrew Sutherland, MIT 18.785 Lecture 7, Remark 7.4.