This note investigates modules having quasi-injective and duo submodules. We introduce a new generalization of C_1-module. The main method that was adopted in this is how to obtain submodule N M the characteristic Quasi-injective. investigate relationship between pseudo-injective module Quasi-injective property Finally, we anti-hopfian module.

In this paper we study an open problem: Is it true that each homotopy domination of a polyhedron over itself is equivalence? Since every ANR has the type polyhedron, question related in obvious way to longstanding Borsuk's problem (1967) two ANR's dominating other have same type? The answer positive for all polyhedra (or ANR's) with polycyclic-by-finite fundamental groups, manifolds, and clearl...

In this paper we defined the concept of module amenability of Banach algebras and module connes amenability of module dual Banach algebras.Also we assert the concept of module Arens regularity that is different with [1] and investigate the relation between module amenability of Banach algebras and connes module amenability of module second dual Banach algebras.In the following we studythe...

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. • If H is a non-parabolic subgroup of a relatively hyperb...

Let $ (A,| cdot |) $ be a real Banach algebra, a complex algebra $ A_mathbb{C} $ be a complexification of $ A $ and $ | | cdot | | $ be an algebra norm on  $ A_mathbb{C}  $  satisfying a simple condition together with the norm $ | cdot | $ on $ A$.  In this paper we first show that $ A^* $ is a real Banach $ A^{**}$-module if and only if $ (A_mathbb{C})^* $ is a complex Banach $ (A_mathbb{C})^{...

We characterize co-Hopfian finitely generated torsion free nilpotent groups in terms of their Lie algebra automorphisms, and construct many examples of such groups.

In this paper we introduce the notion of e-gH modules which is a proper generalization Hopfian and defined as, module called if, any surjective -endomorphism has an e-small kernel, ring e-gH. We give some characterizations properties modules.

We prove that the braid group on 4 or more strands modulo its center is co-Hopfian. We then show that any injective endomorphism of these braid groups is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from the braid group on n strands to the braid group on n + 1 strands is geometric (n ≥ 7). Additionally, we obtain related r...

let $g$ be a group with identity $e$. let $r$ be a $g$-graded commutative ring with a non-zero identity and $m$ be a graded $r$-module. in this article, we introduce the concept of graded almost semiprime submodules. also, we investigate some basic properties of graded almost semiprime and graded weakly semiprime submodules and give some characterizations of them.