الفهرس 
PRELIMINARIESOnly 14 pages are availabe for public view
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Abstract
The study of the relation between rings and both their overrings (extensions) and subrings is a part and parcel of ring theory. The development of the formal study of ring theory has been guided by a huge number of different ring extensions; introduced and investigated for a variety of reasons.
The question of when do certain properties transfer from any ring R to its many types of extensions and vice versa has also been of interest to many algebraists for a long time. A similar question between a module and an overmodule has been pursued in module theory. These questions have been important topics of research and have been crucial in the development of algebra especially of ring and module theory.
It appears that the research work on the wide varieties of extensions is spread throughout the literature in disparate research papers.
The issue of ascertaining how various ring-theoretic concepts behave under certain types of change of rings, such as subrings and ring extensions has always been of fundamental interest among ring theorists. Therefore, the motivation of this thesis is to study the transfer of some algebraic properties between the base ring or module and some of their extensions.
In the present thesis we consider three types of ring and module extensions. The first extension is Ore extensions and the polynomial modules over it. The second extension is the skew generalized power series rings and modules over it. The third extension is the Mal’cev-Neumann series rings and modules over it. Also, we consider different algebraic properties which are shared by the ground rings or modules and the mentioned classes of rings and modules extensions.
The thesis consists of five chapters:
In chapter 1, we provide the preliminaries, definitions and results on the classes of ring extensions used in the subsequent chapters.
In chapter 2, a right R-module M_R is called Hopfian if any surjective endomorphism of M_R is an isomorphism. We investigate Hopfian property on the skew generalized power series module 〖[[M^(S,≤)]]〗_([[R^(S,≤),ω]]).
The results of this chapter are accepted for publication in
”Skew generalized power series Hopfian modules”, Arab J. Math. Sci. [40].
In chapter 3, a right R-module M_R is called a PS-module if its socle, Soc(M_R), is projective. We investigate PS-modules over Ore extension and skew generalized power series extension. Then, under certain conditions, we prove that:
(1) If M_R is a right PS-module, then 〖M[x]〗_(R[x;α,δ]) is a right PS-module.
(2) If M_R is a right PS-module, then 〖[[M^(S,≤)]]〗_([[R^(S,≤),ω]]) is a right PS-module.
The results of this chapter are published in
”PS-modules over Ore extensions and skew generalized power series rings”, Int. J. Math. Math. Sci., (2015), DOI: 10.1155/2015/879129 [39].
In chapter 4, a right R-module M_R is called right zip provided that if the right annihilator of a subset X of M_R is zero, then there exists a finite subset Y⊆X such that r_R (Y)=0. We study the transfer of the right zip property between the right R-module M_R and modules over Ore extension, skew generalized power series extension and Mal’cev-Neumann series extension. Namely, we show under certain conditions that:
(1) M_R is a right zip R-module if and only if 〖M[x]〗_(R[x;α,δ]) is a right zip R[x;α,δ]-module.
(2) M_R is a right zip R-module if and only if 〖[[M^(S,≤)]]〗_([[R^(S,≤),ω]]) is a right zip [[R^(S,≤),ω]]-module.
(3) M_R is a right zip R-module if and only if 〖M((G))〗_(R((G;σ;τ))) is a right zip R((G;σ;τ))-module.
The results of this chapter are published in
”Skew generalized power series zip modules”, Southeast Asian Bull. Math., 38(5), (2014), 693-705 [38].
”Zip property on Mal’cev-Neumann series modules”, Le Matematiche, 70(1), (2015), 115-123, DOI: 10.4418/2015.70.1.9 [43].
In chapter 5, we introduce the notion of rings that satisfy the right weak Beachy-Blair condition and study the relationship between the right weak Beachy-Blair condition of a base ring and some of its extension classes.
The results of this chapter are submitted for publication in
”Extensions of rings satisfy the weak Beachy-Blair condition” [41].
”Mal’cev-Neumann series over rings satisfy the weak Beachy-Blair condition” [42].