The main purpose of this work is to introduce the concept of higher N-derivation and study this concept into 2-torsion free prime ring we proved that:Let R be a prime ring of char. 2, U be a Jordan ideal of R and be a higher N-derivation of R, then , for all u U , r R , n N .

The purpose of this paper is to prove the following result: Let R be a 2-torsion free ring and T: R?R an additive mapping such that T is left (right) Jordan ?-centralizers on R. Then T is a left (right) ?-centralizer of R, if one of the following conditions hold (i) R is a semiprime ring has a commutator which is not a zero divisor . (ii) R is a non commutative prime ring . (iii) R is a commutative semiprime ring, where ? be surjective endomorphism of R . It is also proved that if T(x?y)=T(x)??(y)=?(x)?T(y) for all x, y ? R and ?-centralizers of R coincide under same condition and ?(Z(R)) = Z(R) .

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.

   Let M is a Г-ring. In this paper the concept of orthogonal symmetric higher bi-derivations on semiprime Г-ring is presented and studied and the relations of two symmetric higher bi-derivations on Г-ring are introduced.

      This paper investigates the concept (α, β) derivation on semiring and extend a few results of this map on prime semiring. We establish the commutativity of prime semiring and investigate when (α, β) derivation becomes zero.

We introduced the nomenclature of orthogonal G -m-derivations and orthogonal generalized G -m-derivations in semi-prime G -near-rings and provide a few essentials and enough provision for generalized G -n-derivations in semi-prime G -near-rings by orthogonal.

Let R be a 2-torision free prime ring and ?, ?? Aut(R). Furthermore, G: R×R?R is a symmetric generalized (?, ?)-Biderivation associated with a nonzero (?, ?)-Biderivation D. In this paper some certain identities are presented satisfying by the traces of G and D on an ideal of R which forces R to be commutative

The basic concepts of some near open subgraphs, near rough, near exact and near fuzzy graphs are introduced and sufficiently illustrated. The Gm-closure space induced by closure operators is used to generalize the basic rough graph concepts. We introduce the near exactness and near roughness by applying the near concepts to make more accuracy for definability of graphs. We give a new definition for a membership function to find near interior, near boundary and near exterior vertices. Moreover, proved results, examples and counter examples are provided. The Gm-closure structure which suggested in this paper opens up the way for applying rich amount of topological facts and methods in the process of granular computing.

     Ring theory is one of the influential branches of abstract algebra. In this field, many algebraic problems have been considered by mathematical researchers who are working in this field. However, some new concepts have been created and developed to present some algebraic structures with their properties. Rings with derivations have been studied fifty years ago, especially the relationships between the derivations and the structure of a ring. By using the notatin of derivation, many results have been obtained in the literature with different types of derivations. In this paper, the concept of the derivation theory of a ring has been considered. This study presented the definition of

     Ring theory is one of the influ

The main purpose of this paper is to study some results concerning reduced ring with another concepts as semiprime ring ,prime ring,essential ideal ,derivations and homomorphism ,we give some results a bout that.