FULLY GENERALIZED MODULE

Abstract

This paper introduces the concept of fully generalized modules as an extension of classical module theory over generalized algebraic structures. We prove that every fully generalized module forms a normal generalized group and contains abelian subsets. We further establish the existence of a trivial fully generalized submodule and show that the family of fully generalized submodules is closed under sums and intersections. In addition, we prove the existence of fully generalized module homomorphisms and derive their elementary properties. These results extend existing work on generalized modules and provide a broader theoretical framework for studying module-like structures in generalized algebra. Potential directions for future research include the study of direct sums, fully generalized torsion modules, and fully generalized vector spaces.

Keywords: Generalized group, generalized ring, fully generalized module, submodule, homomorphism..