The main purpose of this work is to generalize Daif's result by introduceing the concept of Jordan (α β permuting 3-derivation on Lie ideal and generalize these result by introducing the concept of generalized Jordan (α β permuting 3-derivation 

In this paper we generalize some of the results due to Bell and Mason on a near-ring N admitting a derivation D , and we will show that the body of evidence on prime near-rings with derivations have the behavior of the ring. Our purpose in this work is to explore further this ring like behavior. Also, we show that under appropriate additional hypothesis a near-ring must be a commutative ring.

In this paper, new concepts which are called: left derivations and generalized left derivations in nearrings have been defined. Furthermore, the commutativity of the 3-prime near-ring which involves some
algebraic identities on generalized left derivation has been studied.

It was known that every left (?,?) -derivation is a Jordan left (?,?) – derivation on ?-prime rings but the converse need not be true. In this paper we give conditions to the converse to be true.

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) .

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.

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

     Let R be an associative ring. The essential purpose of the present paper is to introduce the concept of generalized commuting mapping of R. Let U be a non-empty subset of R, a mapping   : R  R is called a generalized commuting mapping on U if there exist a mapping :R R such that =0, holds for all U. Some results concerning the new concept are presented.

The main purpose of this paper is to investigate some results. When h is  -( ,δ) – Derivation on prime Γ-near-ring G and K is a nonzero semi-group ideal of G, then G is commutative .

In this paper, the concept of Jordan triple higher -homomorphisms on prime

rings is introduced.  A result of Herstein is extended on this concept from the ring  into the prime ring .  We prove that every Jordan triple higher -homomorphism of ring  into prime ring  is either triple higher -homomorphism  or triple higher -anti-homomorphism of  into .