The purpose of this study is to characterize modules V and submodules \(\jmath \) where every submodule N of V not contained in \(\jmath \) is finitely generated. Modules satisfying this condition are called \(\jmath \) -Noetherian modules. Our research explores the application and significance of this concept in the context of various aspects on category of modules. We discuss several properties and characterizations of \(\jmath \) -Noetherian modules. Additionally, we give the Cohen-type and Eakin-Nagata-Formanek theorems of \(\jmath \) -Noetherian module. Furthermore, we examine how these properties extend to various module constructions, including direct sums, exact sequences and polynomial modules. A significant outcome of our research is the discovery and description of distinct modules that characterize our concepts and results. This work not only enriches the theoretical understanding of ring structures but also contributes to the broader field of algebraic theory through practical examples and applications.
