Introduction

Abstract

Tracing back the Nile to its origin must be about as difficult as tracing back the origins of our interest in the theory of orders. At many junctions one has to choose in an almost arbitrary way which is the Nile and which is the other river joining it, wondering whether in such problems one should stick to the wider or to the deeper stream. Perhaps a convenient solution is to recognize that there are many sources and then to list just a few. Those inspired by number theory will certainly think first about the theory of maximal orders over Dedekind domains in number fields, the representation theory-based algebraist will refer to integral group rings, an algebraic geometer will perhaps point to orders over normal domains, and the ring theorist might view orders in central simple algebras as his favorite class of P.I. rings. In these topics graded orders and orders over graded rings appear not only as natural examples, but also as important basic ingredients: crossed products for finite groups, group rings considered as graded rings, orders over projective varieties, rings of generic matrices, trace rings, etc. On these observations we founded our belief that the application of methods from the theory of graded rings to the special case of orders may lead to some interesting topics for research, new points of view, and results. The formulation of this intent alone creates several problems of choice.