The aim of this paper is to study unitary Regular modules on commutative rings with identity. Regularity accompanied by cocyclic property results in some prime-related conclusions on both modules and rings. Further to this, Regularity addresses also radical property of submodules and they are related closely. This property not only affects the modules on ring R but also restricts R to totally idempotent one.

Let M be a finitely generated module over a Regular local ring (R, n), and let M = {Mi} be an n-stable filtration on M. As a consequence of a recent result by Rossi and Sharifan in [14], we prove that if the i-th Betti numbers of M and (grM({M}) coincide with each other, then for each j ³ i the j-th Betti numbers of them are the same and Syzi(M) is a Koszul module provided that Syzi(grM(M) is component wise linear.

We study Bass modules, Bass rings, and related concepts from a model theoretic point of view. We observe that the class of Bass modules (over a fixed ring) is not stable under elementary equivalence. We observe that under which conditions the class of Bass rings are stable under elementary equivalence.

This article is a continuation of E. Momtahan’s work in which he studied the literature via countable injectivity. Along this line, N0-Ikeda-Nakayama rings have been studied and among other things, new characterizations of basically disconnected and extremally disconnected spaces have been given. Furthermore we will observe that a boolean ring is N0-self-injective if and only if it is an N0-IN ring.

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.

Hypergroups that have at least one identity element and where each element has at least one inverse are called Regular hypergroup. In this regards, for a Regular hypergroup $H$, it is shown that there exists a correspondence between the set of all strongly Regular relations on $H$ and the set of all normal subhypergroups of $H$ containing $S_{\beta}$. More precisely, it has been proven that for every strongly Regular relation $\rho$ on $H$, there exists a unique normal subhypergroup of $H$ containing $S_{\beta}$, such that its quotient is a group, isomorphic to $H/\rho$. Furthermore, this correspondence is extended to a lattice isomorphism between them.

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.