On Modules over A Group

Abstract

For a finite group $G$, by the endomorphism ring of a module $M$ over a
commutative ring $R$, we define a structure for $M$ to make it an $RG$%
--module so that we study the relations between the properties of $R$%
--modules and $RG$--modules. Mainly, we prove that $Rad_{R}M$ is an $RG$%
--submodule of $M$ if $M$ is an $RG$--module; also $Rad_{R}M\subseteq
Rad_{RG}M$ where $Rad_{A}M$ is the intersection of the maximal $A$%
--submodule of module $M$ over a ring $A$. We also verify that $M$ is an
injective (projective) $R$--module if and only if $M$ is an injective
(projective) $RG$--module.