In this thesis, the concepts of bounded and fully bounded rings are generalized for modules and they have employed to characterize Rings with essential socle and Artinian, semi-Artinian and pre semi-Artinian Rings. We give an upper bound on the Krull dimension of fully bounded modules using the classical Krull dimension of the base ring. Then, we generalize the concept of the classical Krull dimension of rings for modules and we will prove some basic results. Finally, we will study pure projectivity and pure injectivity of modules over a formal lower triangular matrix rings with applications in general ring theory.
