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.

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 A be a banach ternary algebra over a scalar  eld R or C and X be a ternary banach A{module. Let  ;  and  be linear mappings on A, a linear mapping D: (A; [ ]A)! (X; [ ]X) is called a Lie ternary ( ;  ;  ){derivation, if...

In the present paper, the concepts of module (uni- form) approximate amenability and contractibility of banach al- gebras that are modules over another banach algebra, are introduced. The general theory is developed and some hereditary properties are given. In analogy with the banach algebraic approximate amenability, it is shown that module approximate amenability and contractibility are the same properties. It is also shown that module uniform approximate (contractibility) amenability and module (contractibility, respectively) amenability for commutative banach modules are equivalent. Applying these results to l1 (S) as an l1 (E) -module, for an inverse semigroup S with the set of idem- potents E, it is shown that l1 (S) is module approximately amenable (contractible) if and only if it is module uniformly approximately amenable if and only if S is amenable. Moreover, l1 (S)** is module (uniformly) approximately amenable if and only if a maximal group homomorphic image of S is finite.

We study the topological centers of some specific adjoints of a banach module action. Then, we investigate the Arens regularity and strong irregularity of these actions.

Let A be a banach algebra and X be a banach A-bimodule. In this paper, we define a new product on A ÅX and generalize the module extension banach algebras. We obtain characterizations of Arens regularity, commutativity, semisimplity, and study the ideal structure and derivations of this new banach algebra.