Title: On Modules over a G-set,
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.