On Modules over G-sets

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 ∑_s∈Sm_ss where m_s∈M. 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)_RG. 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)_RG with regard to the properties of M_R, S and G.