الفهرس 
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Abstract
It is undoubtedly that modules and rings are ubiquitous objects of abstract
algebra and they are delight to construct. A natural strategy for
investigating the structure of a ring is to consider a given set of conditions
on the modules it admits. The study of rings and modules would be de -
cient without a connection with homological algebra. In the solar system
of homological algebra the sun is certainly the theory of projective and
injective modules. If R is an associative ring with 1 then every R-module
M can be embedded in an injective R-module. Among the injective modules
that contain M there is a minimal one called the injective hull of M,
denoted by E(M). The injective hull of M is unique up to isomorphism
and in some sense, it is the best approximation of M by an injective module.
The concept of essential extensions will prove to be indispensable in
dealing with uniform dimensions and in the formation of the injective hull of modules which is crucial for the theory of rings of quotients. Injective
modules are closely related to essential extensions. Actually, a module is
injective if and only if it has no proper essential extensions. During the
past twenty years, the theory of injective modules has enjoyed a period of vigorous development. The concept of injectivity has been extended and
characterizations of quasi-Frobenius rings, perfect rings, Kasch rings, and
semiartinian rings in terms of extended injectivities of modules have been
established,[1],[9],[15],[20].
The motivation underlying graded rings and modules is almost contrary
to the one of representation theory. The graded methods are not aiming
to obtain information about the grading group of the graded ring
R = 2GR. On the contrary, the existence of a G-gradation is used to
relate R and Re, e is the unit element of G, or graded to ungraded properties
of R. Graded rings have been proved to be a very satisfactory tool in
the study of algebraic geometry where they are used to gain information
about projective varieties, [21],[10].
This thesis has two predominant objectives. On one hand, we investigate
the relation between the concept of graded essential extensions and graded
injectivity of graded modules. Actually, a characterization of the graded injective hull in terms of graded essential extensions is deduced and graded
versions of (Echman-Schopf) theorem [16] and (Papp-Bass) theorem [4]
has been proved. On the other hand, we study one of the most important
extensions of graded injectivity, namely the graded soc-injectivity. Roughly
speaking, after de ning the graded soc-injectivity of graded modules we
provide characterizations of semiartinian graded rings, graded noetherian
rings, and graded quasi-Frobenius rings in terms of graded soc-injectivity.
The new results we have obtained so far are displayed in the third chapter of this dissertation.
The thesis consists of three chapters.
The rst chapter provides the preliminaries and some background material
to be used in the subsequent chapters. The second chapter consists
of two sections.
The rst section is devoted to the study of graded injective module.
We proved Baer’s theorem for injectivity of graded modules. Actually,
the proof is a slight modi cation of the proof in the ungraded case. The
importance of Baer’s criterion is a clear consequence of its iterated use
throughout the thesis. The relation between injectivity and graded injectivity
is discussed. In the second section we investigate the concept of
H-injective graded modules, where H is a subgroup of the grading group
G, as displayed in [18]. The well-known result of Higman about H-injective
modules over strongly graded rings is presented.The third chapter is divided into two sections.
In the rst section we study basic properties of essential extensions of
graded modules. The graded injective hull of a graded module is characterized
in terms of graded essential extensions. The graded versions of the
well-known Echman-Schopf theorem and Papp-Bass theorem for injective
modules over noetherian rings have been proved.
The second section is devoted to the study of one of the most important
extended injectivity concepts of graded modules which is the graded socinjectivity.
Actually, we introduce and investigate the concepts of graded soc-injectivity and strongly soc-injectivity. We study the relation between
these concepts and the graded semi-simplicity as well as the graded projectivity
of graded modules. We round o by establishing important characterization
of graded semiartinian rings, graded quasi-Frobenius rings, and
graded noetherian rings in terms of graded soc-injectivity of graded modules.
The main results of this thesis are published in
Pure and Apllied Mathematics Journal, Science Publishing
Group (SPG), 2015, Vol. 4, No. 2, 47-51.