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...

We find a relationship between the graded quotients of a filtered holonomic D-module, their duals as coherent sheaves, and the characteristic variety, in case the filtered D-module underlies a polarized Hodge module on a smooth algebraic variety. The proof is based on M. Saito’s result that the associated graded module is Cohen–Macaulay, and on local duality for the cotangent bundle. The result...

We define the derivation module for a homomorphism of inverse semigroups, generalizing a construction for groups due to Crowell. For a presentation map from a free inverse semigroup, we can then define its relation module as the kernel of a canonical map from the derivation module to the augmentation module. The constructions are analogues of the first steps in the Gruenberg resolution obtained...

Here we present a DNA module platform for developing simple aptamer sensors based on a microarray format combined with secondary structure prediction in silico. The platform comprises four parts: (i) aptamer, (ii) joint module, (iii) terminal stem, and (iv) a DNAzyme that catalyzes a redox reaction controlled by a structural change induced by aptamer/target binding. First, we developed a joint ...

For n ≥ 4, we describe Artin’s braid group on n strings as a crossed module over itself. In particular, we interpret the braid relations as crossed module structure relations. Subject classification: Primary: 20F36 57M05 57M20 57M25; Secondary: 18D50 18G55 20C08 55P48