On Generalizations of Injective and Extending Modules
Mahmoud Saad Mehany Mahmoud;
Abstract
Throughout this thesis R will be a ring with identity and all modules considered will be unitary right R-modules.
A module is a CS-module (or an extending, or a module with (C1)) if every submodule is essential in a direct summand (equivalently if every closed submodule is a direct summand). CS-modules generalize injective, and quasi-injective modules. A module M is 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 a direct summand of M. A module M is quasi-continouos if it satisfies (C1) and the following condition (C3): If M1 and M2 are direct summands of M such that M1 M2 = 0, then M1 M2 is a direct summand of M.
The present thesis, which consists of four chapters, introduces some important new aspects in the theory of generalizations of injective modules. These new concepts includes CK-injective modules, CK-jective modules, C-mono-injective, CI-injective.
The first chapter provides the preliminaries and some background results to be used in subsequent chapters, such as basic definitions of modules, indecomposable decompositions, essential, small and closed submodules, some special classes of modules, the socle and the radical of a module, projective and injective modules.
In the second chapter, we investigate modules with the property that every submodule is essential in a direct summand, in literature such modules are called CS-modules.
In the third chapter, we weaken the condition (C1), and study the concept of weak-CS-modules (for short W-CS-modules). Such modules M are defined by: for every submodule A of M, and a summand B of M, there exists a complement K of A B in B which is a summand. We also study the condition (C1*) for a module M: for every submodule A of M, there exists a complement K of A in M which is a summand. By relaxing the condition (C3) (which is given in chapter 2) we have the condition (C3*) for a module M: if A is a closed submodule of M, and B is a summand of M, with A B = 0, then A B is a closed submodule of M.
In the fourth chapter, we introduce the concept of CK-N-injectivity as a generalization of N-injectivity. In Theorem 4.2 we give a homomorphism diagram representation of such concept, as well as an equivalent condition in terms of module decompositions. In Proposition 4.3, we show that if M is CK-N-injective, then M ⊕ N = M ⊕ C holds for every complement C of M in M ⊕ N, with C N ≤c N. In Definition 4.13, we introduce the concept CK-N-jectivity is also dealt with, as a generalization of CK-N-injectivity, and study some of the important properties for direct sums of modules.
A module is a CS-module (or an extending, or a module with (C1)) if every submodule is essential in a direct summand (equivalently if every closed submodule is a direct summand). CS-modules generalize injective, and quasi-injective modules. A module M is 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 a direct summand of M. A module M is quasi-continouos if it satisfies (C1) and the following condition (C3): If M1 and M2 are direct summands of M such that M1 M2 = 0, then M1 M2 is a direct summand of M.
The present thesis, which consists of four chapters, introduces some important new aspects in the theory of generalizations of injective modules. These new concepts includes CK-injective modules, CK-jective modules, C-mono-injective, CI-injective.
The first chapter provides the preliminaries and some background results to be used in subsequent chapters, such as basic definitions of modules, indecomposable decompositions, essential, small and closed submodules, some special classes of modules, the socle and the radical of a module, projective and injective modules.
In the second chapter, we investigate modules with the property that every submodule is essential in a direct summand, in literature such modules are called CS-modules.
In the third chapter, we weaken the condition (C1), and study the concept of weak-CS-modules (for short W-CS-modules). Such modules M are defined by: for every submodule A of M, and a summand B of M, there exists a complement K of A B in B which is a summand. We also study the condition (C1*) for a module M: for every submodule A of M, there exists a complement K of A in M which is a summand. By relaxing the condition (C3) (which is given in chapter 2) we have the condition (C3*) for a module M: if A is a closed submodule of M, and B is a summand of M, with A B = 0, then A B is a closed submodule of M.
In the fourth chapter, we introduce the concept of CK-N-injectivity as a generalization of N-injectivity. In Theorem 4.2 we give a homomorphism diagram representation of such concept, as well as an equivalent condition in terms of module decompositions. In Proposition 4.3, we show that if M is CK-N-injective, then M ⊕ N = M ⊕ C holds for every complement C of M in M ⊕ N, with C N ≤c N. In Definition 4.13, we introduce the concept CK-N-jectivity is also dealt with, as a generalization of CK-N-injectivity, and study some of the important properties for direct sums of modules.
Other data
| Title | On Generalizations of Injective and Extending Modules | Other Titles | عن تعميمات للتشكيلات الحاقنة والتشكيلات الممتدة | Authors | Mahmoud Saad Mehany Mahmoud | Issue Date | 2016 |
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