Let R be a graded ring and M be a graded R -module. We define a topology on graded prime spectrum G- Spec(M) of the graded R -module M which is analogous to that for G- Spec(R), and investigate several properties of the topology.

Let G be a , nitely generated abelian group and M be a G-graded A-module. In general, G-associated prime ideals to M may not exist. In this paper, we introduce the concept of G-attached prime ideals to M as a generalization of G-associated prime ideals which gives a connection between certain G-prime ideals and G-graded modules over a (not necessarily G-graded Noetherian) ring. We prove that the G-attached prime ideals exist for every nonzero G-graded module and this generalization is proper. We transfer many results of G-associated prime ideals to G-attached prime ideals and give some applications of it.

LetG be a group. Let R be a G -graded commutative ring with identity, and let M be a G -graded module over R. Two graded submodulesN and K of graded module M are called graded coprime whenever N+K=M. In this paper, some properties of graded coprime submodules are discussed. For example, we show that ifM is a graded finitely generated module, then two graded submodulesN and K of M are graded coprime if and only if gradM (N) and gradM (K)are graded coprime.

One of the factors that can be the link between our intentions and actions and their external consequences is human agency, which indicates the conscious design and intentional execution of actions by the individual in order to influence future events.Objective and Method: This research with a developmental approach of psychometric method and method 1, examines the psychometric indices of the Human Factor Characteristics Scale using the classical theory of test score measurement and the graduated question-answer theory. The purpose of this study, which included high school students in Tehran, was selected by cluster sampling of 500 people as a sample size and statistical analysis was performed on 481 data. To collect the data, the ion Human Agent Characteristics Scale (2011) was used and the research questions were evaluated using IRTPRO and SPSS software.Results:The assumption of local independence based on Pearson x2 index was established by applying Simjima's calibrated question-answer theory and the assumption of being one-dimensional based on the analysis of multidimensional question-answer theory. Diagnosis parameters with question-answer approach and classical approach Test score Both item 25 approach had the lowest and item 2 had the highest diagnosis parameter. The answer thresholds for all the questions were so far apart that no option was covered by the other option, and the options were independently selected by individuals at intervals of theta. The total scale was calculated with Cronbach's alpha of 0.945, intentionality of 0.894, foresight of 0.780, self-reactivity of 0.871 and rethinking of 0.762. Also, the role of each item in internal consistency was investigated by the loop method, which all questions had a favorable role in internal consistency of this scale. The value of the validity coefficient obtained from the question-answer theory was obtained by marginal method for intentionality 0.92, forethought 0.85, self-reaction 0.91, rethinking 0.83..

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 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.

Let G be a monoid with identity e. In this paper, first we introduce the notions of G-graded hyperrings, graded hyperideals and graded hyperfieldsin the sense of Krasner hyperring R. Also, we define the notion of a greded R-hypermodules and some examples are presented. Then we investigate graded maximal, graded prime and graded primary hyper-ideals of a graded hyperring R. Finally, we study graded maximal, graded prime and graded primary subhypermodules of a graded R-hypermodule M and some interesting results on these concepts are given.