## Some special ideals of commutative rings and some special submodules of modules

Thesis Type: Doctorate

Institution Of The Thesis: Marmara University, Faculty of Arts and Sciences, Mathematics, Turkey

Approval Date: 2019

Thesis Language: Turkish

Student: SUAT KOÇ

Principal Supervisor (For Co-Supervisor Theses): Ünsal Tekir

Abstract:

The aim of this thesis is to define various generalizations of prime ideals and r-ideal and to use them in several ring and module characterizations. R. Mohamadian used this class of ideals to characterize integral domains, quasi regular rings and rings satisfying property (A) by defining r-ideals in commutative rings. Inspired by this notion, in Section 2 of the thesis, r-submodules and sr-submodules which are two different extensions of the concept of r-ideal are introduced. The relations between these two submodules are examined and the characterizations of simple and torsion-free modules are given. In Section 3 of the thesis, the class of n-ideals which is stronger than r-ideals is defined and its relations between some classical ideals such as r-ideal, prime ideal and primary ideal are investigated. Furthermore, the n-ideals are used to characterize von Neumann regular rings and UN-rings defined by Călugăreanu. In Section 4 of the thesis, strongly quasi primary ideals which are between the class of primary ideals and the class of quasi primary ideals are defined. Firstly, the relations of strongly quasi primary ideals with prime, primary, quasi primary, 2-prime ideals are given. Furthermore, divided rings, which are a generalizations of valuation domains are characterized in terms of this class of ideals. Then a subgraph 〖Γ^*〗_U (H) of an ideal based zero divisor graph Γ_U (H) defined by Redmond is studied. Also, the relationships between the algebraic properties of H and graphical properties of 〖Γ^*〗_U (H) are investigated. In the last part of the thesis, n-prime ideal which is a new generalization of prime ideal is given. A new topology on this new class of ideals which is similar to Zariski Topology has been established. Thus, the relationships between the topological properties of this topology and the algebraic properties of the given ring are investigated. Further, quasi-local rings with nil maximal ideal, namely, rings with unique nil maximal ideal are characterized in terms of this class of ideals.