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 study the relation between module amenability, weak module amenability and module approximate amenability of Banach algebra. The notation of amenability of Banach algebras was introduced by B. Johnson in [7]. A Banach algebra A is amenable if every bounded derivation from A into any dual Banach A-bimodule is inner, equivalently if H(A; X) = 0 for any Banach A-bimodule X, where H(A; X) is the first Hochschild co-homology group of A with coefficient in X. Also, a Banach algebra A is weakly amenable if H(A; A) = 0. Bade, Curtis and Dales introduced the notion of weak amenability on Banach algebras in [4]. They considered this concept only for commutative Banach algebras. After that Johnson defined the weak amenability for arbitrary Banach algebras.

Let A be a Banach algebra and E be a Banach A -bimodule then S=A ÅE, 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 D-amenability for these Banach algebras and we show that for discrete inverse semigroup S with the set of idempotents ES, the module extension Banach algebra S=l1 (ES) Å l1 (S) is D-amenable as a l1 (ES) -module if and only if l1 (ES) is amenable as Banach algebra.

In this paper we define a congruence ~ on inverse semigroup S such that amenability of S is equivalent to amenability of S/ ~. We study module amenability of semigroup algebra i1(S/ ~) when S is an inverse semigroup with idempotents E and prove that it is equivalent to module amenability of i1 (S). The main difference of this action with the more studied trivial action is that in this case the corresponding homomorphic image is a Clifford semigroup rather than a discrete group.

In this paper, by considering the notion of extended BCK- module, we define the concepts of free extended BCK-module, free object in category of extended BCK-modules and we state and prove some related results. Specially, we define the notion of idempotent extended BCK-module and we get some important results in free extended BCK- modules. In particular, in category of idempotent extended BCK-modules, we give a method to make a free object on a nonempty set and in BCK- algebra of order 2, we give a method to make a basis for unitary extended BCK-modules. Finally, we define the notions of projective and productive modules and we investigate the relation between free modules and projective modules. In special case, we state the relation between free modules and productive modules.

Let A be a Banach algebra and X be a Banach A-bimodule. Then S = AÅX, the l1-direct sum of A and X becomes a module extension Banach algebra when equipped with the algebra product (a, x).(a’, x’) = (aa’, ax’+xa’). In this paper, we investigate biflatness and biprojectivity for these Banach algebras. We also discuss on automatic continuity of derivations on S = A Å A.

Let s be a linear mapping from a dense subalgebra A of a Banach algebraB into B. In this note, we study the closability of a module s derivation from A into a B- bimodule M. Applying the notions of torsion-free modules and essential ideals, we present several results concerning the closability of such derivations. Also we investigate the closability of module s derivations of the C*- algebra B into a Hilbert B- bimodule M.

Let A be a Banach algebra, A’ ’ a Banach A-module. In this paper, we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions, and them investigate those conditions under which the second adjoint of a derivation into a dual Banach algebra module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included. A^(4) is a banach algebra with four Arens products. The bilinear map T is Arens regular when the equality T*** = T^( r***r ). If T: A × A’ ’ → A’ ’ is multiplication left module on A, the following statements are equivalent, i: T is regular ii: T**** = T^(r****r) iii: T****( A’ ’ ’ , A’ ’ ) ⊆ A’ ’ ’ iv: the linear map a → T*( a’ ’ ’ , a): A → A’ ’ ’ is weakly compact for every a’ ’ ’ ∈ A’ ’ ’ . Also If module actions are regular, then every inner derivation D: A → A’ ’ ’ is weakly compact; moreover, D**: (A’ ’ , □ ) → A^(5) and D**: (A’ ’ , ⋄ ) → A^(5) are also inner derivation.