| This dissertation consists of six chapters. In Chapter 1, we illustrate the background of module's development and the theory of modules play important roles in the process of algebra, and the progress related to supplemented modules, GS-modules and lifting modules. In Chapter 2, some basic definitions and important lemmas related to this paper are given. In Chapter 3, we introduce the concepts of cofinitely GS (WGS)-modules and cofinitely semilocal modules, and discusses some properties of these modules, such as homomorphic images, infinite (direct) sums and equivalent characterizations etal.Meanwhile, As a proper generalization of WGS-modules, the notion of closed WGS-modules are defined and some basic properties of these modules are investigated. In Chapter 4, the concept of S~2-modules are posed as a generalization of modules with the (P~*) condition, and the concepts of completely S~2-modules and WS~2-modules as well as modules with the weakly (P~*) condition are introduced, and related properties of these modules are discussed, some characterizations of (semi-)perfect rings are given in terms of S~2-modules. In Chapter 5, we introduce the concepts ofδ-hollow modules and modules with theδ-(P~*) condition, and use theδ-(P~*) condition gives equivalent characterization ofδ-lifting modules under certain condition, and the decomposition of modules with theδ-(P~*) condition are given. In Chapter 6, dual to the concept of lifting modules, (generalized) extending modules are investigated further. We introduce the concepts of (strongly) FI-generalized extending modules and some properties of these modules are given.
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