Let  be a prime ring,  be a non-zero ideal of  and   be automorphism on. A mapping  is called a multiplicative (generalized)  reverse derivation if  where  is any map (not necessarily additive). In this paper, we proved the commutativity of a prime ring R admitting a multiplicative (generalized)  reverse derivation  satisfying any one of the properties:

 for all x, y  

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

Let R be a Г-ring, and σ, τ be two automorphisms of R. An additive mapping d from a Γ-ring R into itself is called a (σ,τ)-derivation on R if d(aαb) = d(a)α σ(b) + τ(a)αd(b), holds for all a,b ∈R and α∈Γ. d is called strong commutativity preserving (SCP) on R if [d(a), d(b)]_α = [a,b]_α^(σ,τ) holds for all a,b∈R and α∈Γ. In this paper, we investigate the commutativity of R by the strong commutativity preserving (σ,τ)-derivation d satisfied some properties, when R is prime and semi prime Г-ring.

We define a new concept, called " generalized right  -derivation", in near-ring and obtain new essential results in this field. Moreover we improve this paper with examples that show that the assumptions used are necessary.

n this paper , we prove that if T is a 2-torsion free triangular ring and be a family of additive mapping then satisfying is a higher centralizer which is means that is Jordan higher centralizer on 2-torsion free triangular ring if and only if is a higher centralizer and also we prove that if be a family of additive mapping satisfying the relation Σ , Then is a higher centralizer.

The concepts of generalized higher derivations, Jordan generalized higher derivations, and Jordan generalized triple higher derivations on Γ-ring M into ΓM-modules X are presented. We prove that every Jordan generalized higher derivation of Γ-ring M into 2-torsion free ΓM-module X, such that aαbβc=aβbαc, for all a, b, c M and α,βΓ, is Jordan generalized triple higher derivation of M into X.

In this paper, we study the concepts of generalized reverse derivation, Jordan
generalized reverse derivation and Jordan generalized triple reverse derivation on -
ring M. The aim of this paper is to prove that every Jordan generalized reverse
derivation of -ring M is generalized reverse derivation of M.

   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.

In this paper, we provide some types of - -spaces, namely, - ( )- (respectively, - ( )- , - ( )- and - ( )-) spaces for minimal structure spaces which are denoted by ( -spaces). Some properties and examples are given.
The relationships between a number of types of - -spaces and the other existing types of weaker and stronger forms of -spaces are investigated. Finally, new types of open (respectively, closed) functions of -spaces are introduced and some of their properties are studied.

This work generalizes Park and Jung's results by introducing the concept of generalized permuting 3-derivation on Lie ideal.