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 .

In this paper, we will generalized some results related to centralizer concept on
prime and semiprime Γ-rings of characteristic different from 2 .These results
relating to some results concerning left centralizer on Γ-rings.

We present the concept of maps Γ- periodi2 on Γ -near-ring S. Our main goal is to research and explore the presence and mapping traits such as h Γ –hom anti-Γ –hom, Γ –α-derivations of Γ -periodi2 on Γ- near-rings.

Let M be a prime Γ-ring satisfying abc  abc for all a,b,cM and
,  with center Z, and U be a Lie (Jordan) ideal. A mapping d :M M
is called Γ- centralizing if u d u Z  [ , ( )] for all uU and  .In this paper
, we studied Lie and Jordan ideal in a prime Γ - ring M together with Γ -
centralizing derivations on 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 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.

In this paper, we introduce the concept of Jordan  –algebra, special Jordan  –algebra and triple  –homomorphisms. We also introduce Bi -  –derivations and Annihilator of Jordan algebra. Finally, we study the triple  –homomorphisms and Bi -  –derivations on Jordan algebra.