Characterization of modules and rings with the aid of a group

The objective of the thesis is to define a structure for a module over a commutative ring to make it a module over the group ring of a finite group by the endomorphism ring of the module, and to study the relations between the properties of some class of modules over the commutative ring and some class of modules over the group ring by the characterization on modules and rings via the notions defined on the structure. In addition, another objective of the thesis is to determine the characterization of submodules of a group module by a module over a group ring and the decomposition of the group module into its submodules for a finite group. The chapter for the discussion-results where our original studies told consist of three sections.Under the title of "RG--Modules Defined over a Finite Group, Maschke's Theorem", a structure is defined for a module over a commutative ring to make it a module over the group ring of a finite group by the endomorphism ring of the module, and this is the first main purpose of our thesis as told before. Moreover, some examples concerning this structure is given in this chapter. After that, the relations between the radicals of modules, the characterization of injective and projective modules, as modules of not only the ring but also the group ring, is studied by this structure. An alternative proof of Maschke's Theorem is given by the results obtained on the characterization of injective and projective modules. Under the title of "The Socle of the RG--Modules Defined over a Finite Group", the simple submodules and the socle of the modules defined on the structure is investigated as modules of not only the ring but also the group ring. Lastly, the submodule characterization of group modules is studied and the decomposition of group modules into their submodules is determined by adapting the analogous notions and theories used for the decomposition of group rings into their subrings.