Throughout this paper we introduce the notion of coextending module as a dual of the class of extending modules. Various properties of this class of modules are given, and some relationships between these modules and other related modules are introduced.

Let  be a commutative ring with 1 and  be left unitary  . In this paper we introduced and studied concept of semi-small compressible module (a     is said to be semi-small compressible module if  can be embedded in every nonzero semi-small submodule of . Equivalently,  is  semi-small compressible module if there exists a monomorphism  , ,     is said to be semi-small retractable module if  , for every non-zero  semi-small sub module in . Equivalently,  is semi-small retractable if there exists a homomorphism  whenever  .     In this paper we introduce and study the concept of semi-small compressible and semi-small retractable s as a generalization of compressible  and retractable  respectively and give some of their adv

This study has three parts, the first one is the synthesis of a novel Schiff bases by the condensation of guanine or 9-[{2-hydroxyethoxy}methyl]-9H-guanine with variety aldehydes to yield four different bases as follows: (E)-2-((4-nitrobenzylidene)amino)-1,9-dihydro-6H-purin-6-one (S1), (E)-2-((4-methoxybenzylidene)amino)-1,9-dihydro-6H-purin-6-one (S2), (E)-2-((2-hydroxybenzylidene) amino)-9-((2-hydroxy ethoxy)methyl)-1,9-dihydro-6H-purin-6-one (S3), and (E)-2-(((9-((2-hydroxy ethoxy)methyl)-6-oxo-6,9-dihydro-1H-purin-2-yl)imino)methyl)benzoic acid (S4). Then, spectroscopic analyses such as Elemental Analysis, UV/VIS, Mass spectra, FTIR, 1H,13C-NMR were made to recognize these bases. In the second part, the ability of synthesized bases to

Let R be a commutative ring with identity, and let M be a unitary left R-module. M is called special selfgenerator or weak multiplication module if for each cyclic submodule Ra of M (equivalently, for each submodule N of M) there exists a family {fi} of endomorphism of M such that Ra = ∑_i▒f_i (M) (equivalently N = ∑_i▒f_i (M)). In this paper we introduce a class of modules properly contained in selfgenerator modules called special selfgenerator modules, and we study some of properties of these modules.

Let R be a commutative ring with identity, and let M be a unitary left R-module. M is called Z-regular if every cyclic submodule (equivalently every finitely generated) is projective and direct summand. And a module M is F-regular if every submodule of M is pure. In this paper we study a class of modules lies between Z-regular and F-regular module, we call these modules regular modules.

A non-zero module M is called hollow, if every proper submodule of M is small. In this work we introduce a generalization of this type of modules; we call it prime hollow modules. Some main properties of this kind of modules are investigated and the relation between these modules with hollow modules and some other modules are studied, such as semihollow, amply supplemented and lifting modules.

Throughout this paper we introduce the concept of quasi closed submodules which is weaker than the concept of closed submodules. By using this concept we define the class of fully extending modules, where an R-module M is called fully extending if every quasi closed submodule of M is a direct summand.This class of modules is stronger than the class of extending modules. Many results about this concept are given, also many relationships with other related concepts are introduced.

There are two (non-equivalent) generalizations of Von Neuman regular rings to modules; one in the sense of Zelmanowize which is elementwise generalization, and the other in the sense of Fieldhowse. In this work, we introduced and studied the approximately regular modules, as well as many properties and characterizations are considered, also we study the relation between them by using approximately pointwise-projective modules.