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.

      In this paper, we introduce the notion of Jordan generalized Derivation on prime and then some related concepts are discussed. We also verify that every Jordan generalized Derivation is generalized Derivation when  is a 2-torsionfree prime .

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  

We define skew matrix gamma ring and describe the constitution of Jordan left centralizers and derivations on skew matrix gamma ring on a  -ring. We also show the properties of these concepts.

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

In this paper,  we introduce the concepts of  higher reverse left (resp.right)   centralizer, Jordan higher reverse left (resp. right) centralizer, and Jordan triple higher reverse left (resp. right) centralizer of  G-rings. We prove that every Jordan higher reverse left (resp. right) centralizer of a 2-torsion free prime G-ring M is a higher reverse left (resp. right) centralizer of  M.

In this article, the additivity of higher multiplicative mappings, i.e., Jordan mappings, on generalized matrix algebras are studied. Also, the definition of Jordan higher triple product homomorphism is introduced and its additivity on generalized matrix algebras is studied.

In the current paper, we study the structure of Jordan ideals of a 3-prime near-ring which satisfies some algebraic identities involving left generalized derivations and right centralizers. The limitations imposed in the hypothesis were justified by examples.

In this paper, we will prove the following theorem, Let R be a ring with 1 having
a reverse derivation d ≠ 0 such that, for each x R, either d(x) = 0 or d(x) is
invertible in R, then R must be one of the following: (i) a division ring D, (ii) D 2 ,
the ring of 2×2 matrices over D, (iii) D[x]/(x ) 2
where char D = 2, d (D) = 0 and
d(x) = 1 + ax for some a in the center Z of D. Furthermore, if 2R ≠ 0 then R = D 2 is
possible if and only if D does not contain all quadratic extensions of Z, the center of
D.

this paper, we will prove the following theorem, Let R be a ring with 1 having
a reverse derivation d ≠ 0 such that, for each x R, either d(x) = 0 or d(x) is
invertible in R, then R must be one of the following: (i) a division ring D, (ii) D 2 ,
the ring of 2×2 matrices over D, (iii) D[x]/(x ) 2
where char D = 2, d (D) = 0 and
d(x) = 1 + ax for some a in the center Z of D. Furthermore, if 2R ≠ 0 then R = D 2 is
possible if and only if D does not contain all quadratic extensions of Z, the center of
D.