# On Modules over a G-set

## TL;DR

This paper introduces a generalized module construction over a G-set that unifies group algebra and group module theories, providing decomposition methods and conditions for semisimplicity.

## Contribution

It defines a new module MS over the group ring RG for a G-set, unifying existing theories and establishing properties including decomposition and semisimplicity criteria.

## Key findings

- MS can be decomposed into direct sums of RG-submodules.
- Conditions for the semisimplicity of MS are established.
- The theory generalizes and unifies group algebra and group module concepts.

## Abstract

Let R be a commutative ring with unity, M a module over R and let S be a G-set for a finite group G. We define a set MS to be the set of elements expressed as the formal finite sum of the form similar to the elements of group ring RG. The set MS is a module over the group ring RG under the addition and the scalar multiplication similar to the RG-module MG. With this notion, we not only generalize but also unify the theories of both of the group algebra and the group module, and we also establish some significant properties of MS. In particular, we describe a method for decomposing a given RG-module MS as a direct sum of RG-submodules. Furthermore, we prove the semisimplicity problem of MS with regard to the properties of M, S and G.