On The Relation Between Graded Injective Modules and Graded Frobenius Rings

Abstract

An effective way to understand the behaviour of a ring R is to study the various ways in which R acts on its left and right R-modules. Thus, the theory of modules can be expected to be an essential chapter in the theory of rings. Classically, modules were used in the study of represen­ tation theory of groups. In 1950, with the advent of modern homological algebra, the theory of modules has become much broader in scope. The study of projective and injective modules constitutes the backbone of the modern homological theory of modules. Unlike the projective modules, the left regular module RR is not injective in general. Rings for which RR is injective are called left self-injective. The noetherian self-injective rings, commonly known as quasi-Frobenius(or QF) rings, occupy an especially important place in ring theory. The strongly G-graded rings, where G is a finite group, provided the earliest nontrivial examples of QF-rings. A very important motivation in studying QF-rings is the existence of a remarkable perfect duality between finitely generated left and right modules over QF­ rings, cf.[8],[17]. In the past twenty years, the theory of injective modules and quasi-Frobenius rings has enjoyed a period of vigorous development, e.g[1],[9],[17],[22]. The representation theory of finite groups amounts to the study of modules over the group algebra KG of the group G over the field K. The KG-module structure allows to encode precisely most prop­ erties of a G-action on a K-vector space in ring theoretical data,[4]. The philosophy underlying graded ring theory is almost contrary to the one of