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 (R, P) be a Noetherian unique factorization domain (UFD) and M be a finitely generated R -module. Let I (M) be the first nonzero Fitting ideal of M and the order of M, denoted ordR (M), be the largest integern such that I (M) ÍPn. In this paper, we show that if M is a module of order one, then either M is isomorphic with direct sum of a Free module and a cyclic module or M is isomorphic with a special module represented in the text. We also assert some properties of M while ordR(M)=2.

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, we study modules and filters on semihoops. Firstly, we introduce the definition of modules on semihoops and give some examples to illustrate it. Also, we get some significant results related to modules on semihoops. If the semihoop $G$ can generate an Abelian group, then $G$ is a module of any subalgebra $S$ of the semihoop $G$. Then, we use modules and filters to investigate the relationship between modules and semihoops regarding quotient algebras. Secondly, by introducing the definitions of prime submodules and torsion Free modules on semihoops, we explore the relationship among prime modules, filters, and torsion Free modules. Moreover, we discuss the relationship between the images and inverse images under the homomorphism of semihoops and modules, respectively. Finally, we define multiplication modules and comultiplication modules on semihoops. We study the relationship among multiplication modules and submodules on semihoops and provide the condition for comultiplication modules to satisfy the descending chain condition.

Using Free simplicial algebras with given CW−basis, it is shown how to construct a Free or totally Free quasi 3-crossed module on suitable construction data. Quasi 3-crossed complexes are introduced and similar Freeness results are given for these are discussed.