On Goldie*-Supplemented Modules and Their Generalizations

The study of supplemented modules is a central theme in module theory, with deep connections to the structure of rings and categories of modules. This paper provides depth investigation of several recent generalizations of supplementation properties, specifically those defined via the Goldie* torsion theory. We explore the notions of Goldie*-supplemented, totally Goldie*-supplemented, and cofinitely Goldie*-supplemented modules. Furthermore, we introduce and characterize the new concept of essential T-Goldie* lifting modules, establishing its fundamental properties and its relationship with the other mentioned classes. Our approach emphasizes a lattice-theoretic framework, examining the implications of these properties on the lattice of submodules, L(M). We prove several key results, including necessary and sufficient conditions for a module to be totally Goldie*-supplemented, a characterization of cofinitely Goldie*-supplemented modules in terms of their finitely generated submodules, and a fundamental lifting property theorem for essential T-Goldie* modules. Numerous examples are provided to illustrate the concepts. Our results generalize and extend recent work in the literature, contributing to a more profound understanding of the hierarchical structure of module classes beneath the umbrella of Goldie*-supplementation.