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 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.

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.

Let $R$ be a commutative ring with identity and $M$ be an $R$-MODULE. It is shown that the usual lattice $\mathcal{V}(_{R}M)$ of varieties of subMODULEs of $M$ is a distributive lattice. If $M$ is a semisimple $R$-MODULE and the unary operation $^{\prime}$ on $\mathcal{V}(_{R}M)$ is defined by $(V(N))^{\prime}=V(\tilde{N})$, where $M=N\oplus \tilde{N}$, then the lattice $\mathcal{V}(_{R}M)$ with $^{\prime}$ forms a Boolean algebra. In this paper, we examine the properties of certain mappings between $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$, in particular considering when these mappings are lattice homomorphisms. It is shown that if $M$ is a faithful PRIMEFUL $R$-MODULE, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic lattices, and therefore $\mathcal{V}(_{R}M)$ and the lattice $\mathcal{R}(R)$ of radical ideals of $R$ are anti-isomorphic lattices. Moreover, if $R$ is a semisimple ring, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic Boolean algebras, and therefore $\mathcal{V}(_{R}M)$ and $\mathcal{L}(R)$ are anti-isomorphic Boolean algebras.

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.