The characterization of ring and module through prime ideal and prime submodule over a ring with unity

The aim of this thesis is to investigate how the notion of prime ideals and submodules and some other notions defined with the help of prime ideals and submodules are used to determine characterizations of rings and modules and to figure out the relationships between these notions and some known module classes. This thesis consists of four chapters. In the first chapter, we briefly mention the historical process of algebra and notations used in this thesis and express why we study these topics. In the second chapter, we give the definitions and important results of some concepts with respect to ring and module theory used in other chapter. The following chapter completely consists of original studies. In the third chapter, under the title of radical formula, we find some classes of left modules which satisfy the radical formula in a noncommutative ring. We also prove that under a certain condition, a finitely generated module over an HNP-ring, which is the generalization of Dedekind domain, both satisfies the radical formula and can be decomposed into a direct sum of torsion module and CS-module. Under the title of left O-prime ideal, we focus on a one-sided generalization of the concept of prime ideal in a noncommutative ring, which is called a left O-prime ideal. Some of its basic properties are investigated, pointing out both similarities and differences between left O-prime ideals and their commutative counterparts. Mainly, we prove a noncommutative generalization of Cohen's Theorem for left O-prime ideals and that any left ideal in R is the intersection of a finite number of left O-prime ideals of a noncommutative ring R satisfying the ascending chain condition on left O-radical ideals. The section under the title of O-prime submodules is mainly devoted to O-prime submodules, which are not only the module version of left O-prime ideal but also a generalization of prime submodules, and the examination of how the O-prime submodules control the structure of modules. As is well-known, the property that when a submodule P becomes prime in case (P:M) is prime ideal, which is known not to hold in general, is of central importance in the module theory . In this thesis, it is proved that a similar property to the above holds for O-prime submodules. We also investigate the relationships between the intersection of all O-prime submodules and strongly nilpotent elements of a module. Under the title of Zariski subspace topologies associated with ideals, we introduce a complement Zariski topology X_{I} associated with an ideal I of a commutative ring R and obtain some necessary and sufficient algebraic conditions for X_{I} to be quasi-compact space or Noetherian. We also define an ideal N_{I}(0), which is a generalization of radical ideal, to help us find a connection between irreducibility of X_{I} and the primeness of N_{I}(0). Furthermore, we are interested in the relationships between the complement Zariski topologies and ideals of a ring in order to find some algebraic and topological tools which allow us to get some characterizations for both rings and subspaces of Zariski topologies. We show that the Zariski topology is the finite union of irreducible open subsets if and only if the ring R is the sum of some ideals I_{i} and N_{I_{i}}(0) is prime ideal of R. Under the title of the structure of some topologies on a module, we construct some topologies on a module M by using the dual Zariski topology and define a continuous map between these topologies. Then we find a topology on M which is homeomorphic to the dual Zariski topology on a quotient module of M. In the fourth chapter, we summarize the results of this thesis, emphasize its significance in the literature and indicate the subjects thought to be studied after this thesis.