Module Amenability of module dual Banach algebras

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 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.Johnsonin [7]. A Banach algebra A is amenable if every bounded derivation from Ainto 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 isweakly 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.