On Generalizations of Continuous Modules

Abstract

The main objective of this thesis is twofold. First, we aim to introduce the concept of weak large extensions for modules as a general case of the notion of large extensions. Some properties of large extensions hold true for weak large extensions, while some oth- ers need special types of modules (e.g. non-singular modules), and special types of rings. Weak large extending modules (modules in which every WL-closed submodule is a direct summand) are also introduced here. We show that, as of the case of extending modules, the second singular submodule of a WL-extending module splits. Second, we aim to introduce and investigate the concept of relative superfluous injective modules and weak superfluous injective modules. On the other hand we introduce and characterize the concepts of super- fluous extending (for short S -extending) modules, weak superfluous extending (for short WS -extending) modules and superfluous quasi-continuous( for short S -quasi-continuous) modules. We prove that essential properties of injective modules still hold true for su- perfluous injective modules and weak superfluous injective modules. We also establish a convenient characterization for S -extending, WS -extending and S-quasi-continuous mod- ules of non-zero radicals.