Commutative Arithmetical Graded Rings

Abstract

The graded rings encountered in this chapter are the ones that will appear as the centres of the graded orders considered in this book. An order graded by an arbitrary group need not have a graded centre, but when the grading group is abelian this property does hold. Because we exclusively consider orders over domains it makes sense to restrict attention to commutative rings which are graded by torsion free abelian groups, in particular where Krull domains are concerned. On the other hand, constructions over gr-Dedekind rings appear as examples or in the constructive methods for studying class groups, hence it will be sufficient to develop the basic facts about gr-Dedekind rings and the related valuation theory in the ℤ-graded case only.