الفهرس 
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Abstract
A module M is called A–injective module if every submodule X of A, any homomorphism φ: X → M can be extended to a homomorphism y :A→ M.
A module M is called injective module if M is A–injective for every module A.
A module M is quasi – injective if M is M–injective.
A module M is a CS–module (or an extending, or a module with (C1)) if every submodule of M is essential in a direct summand (equivalently if every closed submodule is a direct summand, of M).
A module M is called continuous if it satisfies (C1) and the following condition (C2): if a submodule A of M is isomorphic to a direct summand of M, then A is itself a direct summand of M.
A module M is called quasi–continuous if it satisfies (C1) and the following condition (C3): if M1 and M2 are dirct summands of M such that M1 Ç M2 = 0, then M1 Å M2 is a direct summand of M.
A module M is called min–CS–module if every simple submodule of M is essential in a direct summand.
A module M is called 1–CS–module if every uniform submodule of M is essential in a direct summand (or equivalently if every uniform closed submodule is a direct summand).
CS–modules are generalization of (quasi–) continuous modules, which, in turn generalization of (quasi–) injective modules.
CS–modules have been investigated in a large number of papers. M. Okado [39] proved that, over a right noetherian ring, any CS–module is a direct sum of uniform submodules.
M. Kamal [22] studied modules in which complements are summands. In [22] Kamal has given a full characterization of CS–modules over integral domains.
M. Kamal and B. Mueller [20] and [21] have generalized some of the results in [22] to arbitrary rings, and discussed the structure of CS–modules over noetherian rings.
Also M. Kamal [23], [24], and [25] studied the decompositions and direct sums of CS–modules.
M. Kamal [25], and Harmanci and Smith [17] independently proved that a finite direct sum of mutually injective CS–modules is CS.
M. Harada and K. Oshiro [13] considered modules with extending properties, which are closely related to CS.
S. Mohamed and T. Bouhy [34] studied continuous modules and showed that every (quasi–) injective module is continuous.
B. Mueller and S. T. Rizvi [35] and [36] studied continuous and quasi–continuous modules.
V. Camillo, W. K. Nicholson, and M. F. Yousif [2] have dealt with Ikeada- Nakayama rings that are related to continuous and quasi–continuous rings.
For a bimodule SMR , R. Wisbauer, M. F. Yousif, and Y. Zhou [41] have studied an annihilator condition for a module M, namely LS (A Ç B) = LS(A) + LS(B) for any submodules A, B of MR. As a result, they obtained new characterizations of quasi–continuous modules.
W. K. Nicholson and M. F. Yousif generalized self injective rings to principally injective rings. In [37] they have given equivalent conditions for principally injective rings in view of such an annihilator condition. They have also generalized principally injective rings to mininjective rings. In [38] they have given equivalent conditions of mininjective rings in terms of the given annihilator condition.
The present thesis, which consists of four chapters, introduces some important new aspects in regard of generalizations of injective modules.
These new concepts includes (min–CS)*–modules, the equivalent conditions for uniform injective and 1–CS–modules, and the equivalent conditions for P–CS–modules.
The first chapter provides the preliminaries and some background results to be used in subsequent chapters, such as basic definitions of modules, annihilator ideals and annihilator submodules, the chain conditions, indecomposable decompositions, essential, small and closed submodules, the uniform dimension of a module, semisimple modules, the socle and the radical of a module, projective and injective modules.
The second chapter, we investigate modules with the property that every simple submodule is essential in a direct summand; in literature such modules are called min–CS– modules.
In this chapter we also introduce the concept of ES-closed submodule of a module M. In fact we mean by a ES-closed that a submodule C which contains essentially simple submodule.
In the first part of this chapter, we recall the different characterizations of CS-modules, which give rise to the following generalization
(C1*): If A ≤ M, then there is a decomposition M = M1 M2 such that A ∩ M2 = 0, and A M2 ≤e M. It is clear that every CS-module must satisfy the condition (C1*)
One of the aims of this chapter is to investigate the equivalent condition for (min–CS) modules.
We weaken the condition (min–CS) and introduce the concept of (min–CS)* –modules. Such modules M are defined by:
For every simple submodule A of M there exists a complement K of A in M, which is a direct summand.
We also give the necessary and sufficient condition for a direct summand N of a module M to be (min–CS)* module (Proposition 2.34).
In Lemma 2.35 we show that direct sums of two (min–CS)* –modules are (min–CS)* –modules.
Corollary 2.36 indicates the necessary and sufficient condition for a finite direct sum of modules to be (min–CS)* –modules.
In Proposition 2.37 we show that any indecomposable (min–CS)*–module M is uniform.
In the third chapter, we have different characterizations of mininjective modules.
Recall that: a module M is called uniform injective if for every homomorphism from a uniform submodule U of M can be extend to M.
In the second part of this chapter, we have different characterizations of uniform injective modules.
Finally, we investigate modules with the property that every uniform submodule is essential in a direct summand; in literature such modules are called 1-CS-modules.
Theorem 3.23 gives different equivalent conditions for the defining conditions of 1-CS-modules.
Lemma 3.26 gives that the condition (1–CS) is inherited by summands.
Proposition 3.28 gives the condition that when a direct sum of two 1-CS-modules to be 1-CS-module.
In the fourth chapter, Recall that a right R–module MR is called principally injective (for short P–injective) if every homomorphism from a principal right ideal of R to M can be extends to R.
M. A. Kamal, and O. A. Elmnophy [26] studied principally injective modules. They also studied modules with the property that every cyclic submodule is essential in a direct summand. They named such modules are called P–extending modules.
One of the aims of this chapter is to give new characterizations of P–extending modules.
In the second section we have studied bimodules SMR with the following annihilator condition S = Ls (A) + Ls (B), where B is a complement of a simple submodule S of MR, and A is a closure of S in MR. This annihilator condition has a direct connection with the defining condition for min-CS modules for MR. We make use of this to giv a new characterization of min-CS-modules.
In this chapter, we also study bimodules SMR with the following annihilator condition: S = LS (A) + LS (B) for any EC-closed submodule A, where B is a complement of A, in MR. Such annihilator condition has a direct connection with the defining condition of P-extending modules for MR. We make use of this to give a new characterizations of P-extending modules. In fact (Lemma 4.10) and (Corollary 4.11) give the annihilator condition which is in connection with P–extending modules.
Proposition 4.12 and Proposition 4.13 give a new characterization of P–extending modules.
We also study bimodules SMR which satisfy the following condition (ANN):
For every EC-closed submodule A of MR there exists a complement B of A in MR such that S = LS (A) + LS (B). Such modules are P-quasi-continuous modules.
Lemma 4.14 we give a necessary and sufficient condition for P-quasi-continuous modules to satisfy the condition (ANN).
Finally, we introduce the concept of W-EC-modules, namely bimodules SMR for which A = rMLS(A) for every EC-closed submodule A of MR. We also discuss the relation between W-EC-modules and P-(quasi-) continuous modules, for bimodules SMR.
In my opinion, the following results are new: [Theorem 2.30, Lemma 2.32, Proposition 2.34, Lemma 2.35, Corollary 2.36, Proposition 2.37, Proposition 3.18, Lemma 3.19, Proposition 3.20, Theorem 3.21, Theorem 3.23, Lemma 3.24, Lemma 3.25, Lemma 3.26, Proposition 3.28, Lemma 4.9, Corollary 4.10, Proposition 4.11, Proposition 4.12, Lemma 4.13, Lemma 4.14, Proposition 4.15, Lemma 4.16, Proposition 4.17, Proposition 4.18, and Proposition 4.19].
All rings have are considered with unities, and not necessary commutative. All modules here are right R- modules. All items, such as Theorems, Propositions, Lemmas, Corollaries, Remarks and Definitions, are numbered consecutively, (m.n), where m indicates the chapter and n indicates the place of the item within this chapter.