In this work, we introduce a new generalization of both Rationally extending and Goldie extending which is Goldie Rationally extending module which is known as follows: if for any submodule K of an R-module M there is a direct summand U of M (denoted by  U⊆_⊕ M) such that K β_r  U. A β_r  is a relation of K⊆M and U⊆M, which defined as  K β_r  U if and only if  K ⋂U⊆_r K and K⋂U⊆_r U.

In this paper, we introduce a type of modules, namely S-K-nonsingular modules, which is a generalization of K-nonsingular modules. A comprehensive study of these classes of modules is given.

Coupling reaction of 2-amino benzoic acid with the 8-hydroxy quinoline gave the azo ligand (H2L): 5-(2-benzoic acid azo )-8-hydroxy quinoline.Treatment of this ligand with some metal ions (CoII, NiII and CuII ) in ethanolic medium with a (1:2) (M:L) ratio yielded a series of neutral complexes with general Formula[M(HL)2],where: M=Co(II), Ni(II) and Cu(II), HL=anion azo ligand (-1).The prepared complexes were characterized using flame atomic absorption,FT-IR and UV-Vis spectroscopic methods as well as magnetic susceptibility and conductivity measurements.

   In this work, the(m-phenylenediamine) and (2-naphthol) have been used in the synthesis of tetradentate ligand [m-phenylenedi(azo-2-naphthol)][H2L] type (N2O2). The ligand was refluxed in the ethanol with the metal ions [Co(II), Cu(II) and Zn(II)] salts, using triethyleamine as a base in (2:2) molar ratio to give the binuclear  complexes. These complexes were characterised by (A.A), F.T.I.R, (U.V-Vis) spectroscopies, along with conductivity, chloride content and melting point measurement. These studies revealed an octahedral geometries for  Co(II), Cu(II) and Zn(II) complexes with the general structure [M2(L)2(H2O)4]. The ligand and its complexes exhibited biological activity against the Bacillus(G+) strain and the

On Goldie 

Let R be a ring with identity and let M be a left R-module. M is called µ-lifting modulei f for every sub module A of M, There exists a direct summand D of M such that M = D D', for some sub module D' of M such that A≤D and A D'<<_µ D'. The aim of this paper is to introduce properties of µ-lifting modules. Especially, we give characterizations of µ-lifting modules. On the other hand, the notion of amply µ-supplemented iis studied as a generalization of amply supplemented modules, we show that if M is amply µ-supplemented such that every µ-supplement sub module of M

The complexes of the 2-hydroxy-4-Nitro phenyl piperonalidene with metal ions Cr(III), Ni(II), Pt(IV) and Zn(II) were prepared in ethanolic solution. These complexes were characterized by spectroscopic methods, conductivity, metal analyses and magnetic moment measurements. The nature of the complexes formed in ethanolic solution was study following the molar ratio method. From the spectral studies, monomer structures proposed for the nickel (II) and Zinc (II) complexes while dimeric structures for the chromium (III) and platinum (IV) were proposed. Octahedral geometry was suggested for all prepared complexes except zinc (II) has tetrahedral geometry, Structural geometries of these compounds were also suggested in gas phase by using

he concept of small monoform module was introduced by Hadi and Marhun, where a module U is called small monoform if for each non-zero submodule V of U and for every non-zero homomorphism f ∈ Hom R (V, U), implies that ker f is small submodule of V. In this paper the author dualizes this concept; she calls it co-small monoform module. Many fundamental properties of co-small monoform module are given. Partial characterization of co-small monoform module is established. Also, the author dualizes the concept of small quasi-Dedekind modules which given by Hadi and Ghawi. She show that co-small monoform is contained properly in the class of the dual of small quasi-Dedekind modules. Furthermore, some subclasses of co-small monoform are investiga

In this paper, we introduce a new concept named St-polyform modules, and show that the class of St-polyform modules is contained properly in the well-known classes; polyform, strongly essentially quasi-Dedekind and ?-nonsingular modules. Various properties of such modules are obtained. Another characterization of St-polyform module is given. An existence of St-polyform submodules in certain class of modules is considered. The relationships of St-polyform with some related concepts are investigated. Furthermore, we introduce other new classes which are; St-semisimple and ?-non St-singular modules, and we verify that the class of St-polyform modules lies between them.

Gangyong Lee, S.Tariq Rizvi, and Cosmin S.Roman studied Rickart modules.

The main purpose of this paper is to develop the properties of Rickart modules .

We prove that each injective and prime module is a Rickart module. And we give characterizations of some kind of rings in term of Rickart modules.