Extending modules relative to module classes

The purpose of this study is to give an up-to-date presentation of known and new results on extending modules and related notions with respect to an R-module class X. By assuming basic facts from Module Theory, our treatment is essentially self-contained. In the first chapter, some background material is given together with the definitions of the two types of extending module with respect to a class of modules. We investigate the extending property with respect to related module classes and direct sum decompositions of extending modules. We also define two types of weak extending module and compare with extending modules both with respect to a class of modules. The second chapter concerns the structure and properties of extending modules with respect to certain standard classes of modules, namely Goldie torsion modules, non-singular modules, modules with finite uniform dimension and finitely generated modules. We also investigate the particular case of torsion modules over Dedekind domains. The importance of injective modules in Module Theory and more generally in Algebra is obvious in the 1960s and 1970s, largely, but not exclusively, through the impact of the publication of the lecture notes of Carl Faith [9]. Since that time there has been continuing interest in such modules and their various generalizations which arose not only directly from the study of injectives but also from the work of John von Neumann mentioned above. Some results obtained for injective modules can be transferred readily to injective modules with respect to R-module classes X. In chapter three, we investigate the injective and also quasi-injective modules with respect to R-module classes and characterise them