Determination of characterizations of modules and rings with the aid of prime and coprime submodules

The aim of this thesis is to investigate how the notion of coprime submodules and some other notions defined with the help of coprime submodules are used to determine characterizations of rings and modules and to figure out how these notions effect on some known module classes. In the first chapter, some preliminary information concerning ring and module theory which will be used in the subsequent chapters is given. In the second chapter, some results concerning coprime submodules in the literature are given. It is organized in a way that after the second chapter, the three chapters are completely original. In the third chapter, firstly, some characterizations of coprime modules over noncommutative rings are given and some relationships with the different module classes are investigated. Then, by using coprime quotient modules of a module, a number of results concerning attached primes of this module are obtained. Finally, some relationships between the notion of coprime module and the notion of hollow dimension are investigated and a number of results concerning modules with the max property are given by using the notion of coprime module. In the fourth chapter, the notion of coprime radical is dealt with and some characterizations for rings and modules are obtained by using this notion. In this chapter, the notion of m^{∗}-system which is the dual notion of m-systems in rings for modules is come up with and coprime radical of a submodule is characterized via m^{∗}-system sets. In addition to this, the question that in which cases the coprime radical of a module is equal to the socle of this module is dealt with and the rings over which every module satisfies this equality are investigated. Furthermore, some characterizations of simple rings and right Artinian rings are given by using coprime radicals of certain modules. In the fifth chapter, the notions of graded coprime and graded coprimary submodules over graded rings are defined and some characterizations of these submodules are given. It is also proved that every graded injective module has a graded secondary representation over a commutative graded Noetherian ring.