The Category G-GrR-Mod and Group Factorization

In this work, we use the concept of G-weak graded rings and G-weak graded modules, which are based on grading by a set G of left coset representatives for the left action of a subgroup H of a finite X on X, to define the conjugation action of the set G and to generalize and prove some results from the literature. In particular, we prove that a G-weak graded ring R is strongly graded if and only if each G-weak graded R-module V is induced by an ReG-module. Moreover, we prove that the additive induction functor (−)R and the restriction functor (−)eG form an equivalence between the categories G-GrR-Mod and ReG-Mod when R is strongly G-weak graded. Furthermore, some related results and illustrative examples of G-weak graded R-modules and their morphisms are provided.