Generalized Maximal Orders

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.