A gamma T_ pure sub-module also the intersection property for gamma T_pure sub-modules have been studied in this action. Different descriptions and discuss some ownership, as Γ-module Z owns the TΓ_pure intersection property if and only if (J2 ΓK ∩ J^2  ΓF)=J^2 Γ(K ∩ F) for each Γ-ideal J and for all TΓ_pure K, and F in Z Q/P is TΓ_pure sub-module in Z/P, if P in Q.

        Let R be a commutative ring with identity, and let M be a unitary R-module. We introduce a concept of almost bounded submodules as follows: A submodule N of an R-module M is called an almost bounded submodule if there exists xÃŽM, xÏN such that ann_R(N)=ann_R(x).

        In this paper, some properties of almost bounded submodules are given. Also, various basic results about almost bounded submodules are considered.

        Moreover, some relations between almost bounded submodules and other types of modules are considered.

Throughout this paper S will be denote a monoids with zero. In this paper, we introduce the concept of En- prime subact, where a proper subact B of a right S- act As is called En- prime subact if for any endomorphism f of As and a As with f(a)S⊆ Bimplies that either a B or f(As) ⊆ B. The right S-act As is called En-prime if the zero subact of As is En-prime subact. Some various properties of En-prime subact are considered, and also we study some relationships between En-prime subact and some other concepts such as prime subact and maximal subact. 

In this paper, we introduce the concept of almost Quasi-Frobcnius fuzzy ring as a " " of Quasi-Frobenius ring. We give some properties about this concept with qoutient fuzzy ring. Also, we study the fuzzy external direct sum of fuzzy rings.

Let  be a commutative ring with 1 and  be left unitary  . In this paper we introduced and studied concept of semi-small compressible module (a     is said to be semi-small compressible module if  can be embedded in every nonzero semi-small submodule of . Equivalently,  is  semi-small compressible module if there exists a monomorphism  , ,     is said to be semi-small retractable module if  , for every non-zero  semi-small sub module in . Equivalently,  is semi-small retractable if there exists a homomorphism  whenever  .

    In this paper we introduce and study the concept of semi-small compressible and semi-small retractable s as a generalization of compressible  and retractable  respectively and give some of

Our active aim in this paper is to prove the following Let Ŕ be a ring having an
idempotent element e(e  0,e 1) . Suppose that R is a subring of Ŕ which
satisfies:
(i) eR  R and Re  R .
(ii) xR  0 implies x  0 .
(iii ) eRx  0 implies x  0( and hence Rx  0 implies x  0) .
(iv) exeR(1 e)  0 implies exe  0 .
If D is a derivable map of R satisfying D(R )  R ;i, j 1,2. ij ij Then D is
additive. This extend Daif's result to the case R need not contain any non-zero
idempotent element.

In this paper, we proved that if R is a prime ring, U be a nonzero Lie ideal of R , d be a nonzero (?,?)-derivation of R. Then if Ua?Z(R) (or aU?Z(R)) for a?R, then either or U is commutative Also, we assumed that Uis a ring to prove that: (i) If Ua?Z(R) (or aU?Z(R)) for a?R, then either a=0 or U is commutative. (ii) If ad(U)=0 (or d(U)a=0) for a?R, then either a=0 or U is commutative. (iii) If d is a homomorphism on U such that ad(U) ?Z(R)(or d(U)a?Z(R), then a=0 or U is commutative.

The definition of orthogonal generalized higher k-derivation is examined in this paper and we introduced some of its related results.

Number theorists believe that primes play a central role in Number theory and that solving problems related to primes could lead to the resolution of many other unsolved conjectures, including the prime k-tuples conjecture. This paper aims to demonstrate the existence of this conjecture for admissible k-tuples in a positive proportion. The authors achieved this by refining the methods of “Goldston, Pintz and Yildirim” and “James Maynard” for studying bounded gaps between primes and prime k-tuples. These refinements enabled to overcome the previous limitations and restrictions and to show that for a positive proportion of admissible k-tuples, there is the existence of the prime k-tuples conjecture holding for each “k”. The sig

Let M be an R-module. In this paper we introduce the concept of quasi-fully cancellation modules as a generalization of fully cancellation modules. We give the basic properties, several characterizations about this concept. Also, the direct sum and the localization of quasi-fully cancellation modules are studied.