In this paper a Г-ring M is presented. We will study the concept of orthogonal generalized symmetric higher bi-derivations on Г-ring. We prove that if M is a 2-torsion free semiprime    Г-ring ,  and  are orthogonal generalized symmetric higher bi-derivations  associated with symmetric higher bi-derivations   respectively for all n ϵN.

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

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.

 The current paper studied the concept of right n-derivation satisfying certified conditions on semigroup ideals of near-rings and some related properties. Interesting results have been reached, the most prominent of which are the following: Let M be a 3-prime left near-ring and A_1,A_2,…,A_n are nonzero semigroup ideals of M, if d is a right n-derivation of M satisfies on of the following conditions,
d(u_1,u_2,…,(u_j,v_j ),…,u_n )=0 ∀ 〖 u〗_1 〖ϵA〗_1 ,u_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n ϵA〗_u;
d((u_1,v_1 ),(u_2,v_2 ),…,(u_j,v_j ),…,(u_n,v_n ))=0 ∀u_1,v_1 〖ϵA〗_1,u_2,v_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n,v_n ϵA〗_u ;
d((u_1,v_1 ),(u_2,v_2 ),…,(u_j,v_j ),…,(u_n,v_n ))=(u_

      In this study, we prove that let N be a fixed positive integer and R be a semiprime -ring with extended centroid . Suppose that additive maps  such that  is onto,  satisfy one of the following conditions  belong to Г-N- generalized strong commutativity preserving for short; (Γ-N-GSCP) on R   belong to Г-N-anti-generalized strong commutativity preserving for short; (Γ-N-AGSCP)  Then there exists an element  and  additive maps  such that  is of the form  and   when condition (i) is satisfied, and     when condition (ii) is satisfied   

Let M be a weak Nobusawa -ring and γ be a non-zero element of Γ. In this paper, we introduce concept of k-reverse derivation, Jordan k-reverse derivation, generalized k-reverse derivation, and Jordan generalized k-reverse derivation of Γ-ring, and γ-homomorphism, anti-γ-homomorphism of M. Also, we give some commutattivity conditions on γ-prime Γ-ring and γ-semiprime Γ-ring .

Let M be ,-ring and X be ,M-module, Bresar and Vukman studied orthogonal
derivations on semiprime rings. Ashraf and Jamal defined the orthogonal derivations
on -rings M. This research defines and studies the concepts of orthogonal
derivation and orthogonal generalized derivations on ,M -Module X and introduces
the relation between the products of generalized derivations and orthogonality on
,M -module.

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.

The concepts of higher Bi- homomorphism and Jordan higher Bi- homomorphism have been introduced and studied the relation between Jordan and ordinary higher Bi- homomorphism also the concepts of Co- higher Bi- homomorphism and  Co- Jordan higher Bi- homomorphism introduced  and the relation between them in Banach algebra have also been studied.