let a be a banach algebra and e be a banach a-bimodule then s = a  e, the l1-direct sum of a and e becomes a module extension banach algebra when equipped with the algebras product (a; x):(a′; x′) = (aa′; a:x′ + x:a′). in this paper, we investigate △-amenability for these banach algebras and we show that for discrete inverse semigroup s with the set of idempotents es, the module extension bana...

let $r$ be a commutative ring with identity and $m$ be a finitely generated unital $r$-module. in this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. moreover, we investigate some conditions which imply that the module $m$ is the direct sum of some cyclic modules and free modules. then some properties of fitting ideals of...

a module $m$ is called $emph{h}$-cofinitely supplemented if for every cofinite submodule $e$ (i.e. $m/e$ is finitely generated) of $m$ there exists a direct summand $d$ of $m$ such that $m = e + x$ holds if and only if $m = d + x$, for every submodule $x$ of $m$. in this paper we study factors, direct summands and direct sums of $emph{h}$-cofinitely supplemented modules. let $m$ be an $emph{h}$...

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...

Let Jl consist of the groups GÍL m) = (a, b; a~ b a = b"1) where |/.| 4 1 4 |zzi|, lm40 and l, m are coprime. We characterize the endomorphisms of these groups, compute the centralizers of special elements and show that the endomorphism a —• a, b —' b is onto with a nontrivial fully invariant kernel. Hence G(l, m) is non-Hopfian in the'fully invariant sense.' Our purpose is to prove the results...

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples surfaces without boundary whose (pure) are not co-Hopfian; these first such endomorphisms that fail to be surjective. then prove that, subject some topological conditions on the domain surface, any continuous homomorphism (arbitrary) sends Dehn twists is i...

The Baumslag-Solitar group $BS(2,3)$, is a so-called non-Hopfian group, meaning that it has an epimorphism $\phi$ onto itself, not injective. In particular this equivalent to saying $BS(2,3)$ non-trivial quotient isomorphic itself. As consequence the Cayley graph of itself up change generators. We describe on graph-level and take closer look at most common $\phi$. show its kernel free infinite ...

we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...

in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.