Restrict Nearly Quasi Prime (R.S.Q.P) Submodule

Abstract

In this work, we developed a new definition based on a primary sub-module. This sub-module served as the foundation upon which we constructed and analyzed this definition. We examined some of its characteristics, as well as characteristics closely related to this work, and established the conditions and fundamental principles governing the relationships between this definition and previous ones. We also presented some relationships and inverse relationships between a set of previous definitions and the current definition, providing counter examples of relationships that cannot be achieved. Furthermore, we explored the relationship between two sub-modules. We also clarified the relationships between sub-modules when one is contained within another and examined the appropriate conditions that must be imposed on the modules to achieve specific characteristics. We used an important and fundamental condition in module theory, which is the distribution condition, to achieve some important results in the current work. We explained that the intersection operation between submodules is not necessarily closed or guaranteed.