The Injective Profile of a Ring and its Effect on the Structure of Rings

Abstract

This thesis is partly based on recent developments on the subject of what is known as “injectivity domains” in the theory of modules over rings with identity. The subject was suggested as a measurement of how far a module is away from being injective and has gained increased interest over the last few years from people studying rings towards homological properties. The aim of this thesis is to present significant achievements with a unifying approach. Our thesis is primarily concerned with the investigation of a particular class of rings, called rings with no middle class, which is defined by means of injectivity domains. This thesis consists of four chapter. The first chapter contains motivation and historical background of the subject of this thesis. In the second chapter, we give some necessary background material and classifications of some rings by their homological properties to better understand next chapters. In the third chapter, we introduce the notion of injectivity domains and that of poor modules, defined in terms of injectivity domains. The last chapter is concerned with the rings without a middle class. We give a number of properties of these rings and characterize them with respect to hereditary pretorsion classes. We also explore decomposability of rings with no middle class and obtain, in an incisive way, that they can decompose into the product of an indecomposable ring and a semisimple Artinian ring. Finally, we investigate commutative rings without a middle class.