Generalized Maximal Orders

TL;DR

This paper generalizes the concept of maximal orders over algebras by relaxing the condition that the base field must be central, enabling their application to crystalline graded rings.

Contribution

It introduces a generalized definition of maximal orders that applies to crystalline graded rings, extending classical theory beyond central field assumptions.

Findings

Successfully generalized maximal orders to non-central base fields.

Retained key properties of maximal orders in the new setting.

Extended applicability to crystalline graded rings.

Abstract

Maximal Orders over an algebra are a generalization of the concept of a Dedekind domain. The definition given in Maximal Orders by Reiner, assumes that the field over which the algebra is defined is in the center of the order. Since we want to define maximal orders over a Crystalline Graded Ring (defined in Nauwelaerts, E.; Van Oystaeyen, F., Introducing crystalline graded algebras, Algebras and Representation Theory vol 11(2008), no. 2, 133--148.), this concept needs to be generalized. In this paper, we will weaken the condition that the field needs to be in the center, and still retain many of the desired properties of a maximal order.