A Study Of Torsion Theories

This paper is a study of torsion theories and related topics such as pure and divisible for both integral domains and non-integral domains. Motivating the study was a super goal of investigating the existence and uniqueness of torsion-free covering modules over not necessarily integral domains. These are shown to exist and uniquely in the first section of the paper for the integral domain case. Two torsion theories for not necessarily integral domains are studied in the second and third sections.