Compactness in categories and its application in different categories

Abstract

In the paper [HSS] Herrlich, Salicrup and Strecker were able to show that Kuratowski / Mrowka's Theorem concerning compactness for topological spaces could be applied to a wider setting. In this dissertation, which is based on the paper [F subscript 1], we interpret Kuratowski / Mrowka's result in the category R-Mod. Chapter One deals mainly with the preliminary definitions and results and we also show that there is a 1-1 correspondence between torsion theories and standard factorisation systems. In Chapter Two we, obtain for every torsion theory T, a theory of T-compactness which is an extension of the definition of compactness found in [HSS]. We then obtain a characterisation of T-compactness under certain conditions on the ring R and torsion theory T. In Chapter Three we examine the class of T-compact R-modules more closely when the ring R is T-hereditary and T-noetherian. We also obtain further characterisation of T-compactness under these additional conditions. In Chapter Four we show that many topological results have analogues in R-Mod.