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.