Jordan Generalized Centralizerhomo on Prime Rings

In this article we introduce the concepts of generalized centralizerhomo and Jordan generalized centralizerhomo on prime ring. The aim object of this paper we prove: Every Jordan generalized centralizerhomo of 2-torsion free prime ring R is Jordan generalized triple centralizerhomo on R .


Introduction
The objective of the definition of generalized left centralizerhomo is to generalize the definition of left centralizerhomo on rings and to specify the relation between two concepts: generalized left centralizerhomo and Jordan generalized left centralizerhomo with in certain conditions. The definition of prime ring and semiprime ring was presented in [1]. The definition of 2 − torsion free was presented in [2] while the definition of centralizerhomo was presented in [3].On the other hand, [3] was first to introduce the concepts of left centralizerhomo, Jordan left centralizerhomo and generalization on rings, many results were found by the researchers, one of these results was that "every Jordan left centralizer of a 2 − torsion free prime ring R is a centralizer". For more information of left centralizer and homomorphism see [4,5,6,7] One important question can be answered in this paper whether there is a relation between a concept of Jordan generalized left centralizerhomo and a concept of Jordan triple generalized left centralizerhomo within certain conditions.
In this paper we refer to the left centralizerhomo by centralizerhomo for short.

2.
Generalized Centralizerhomo on Prime Rings. Definition 2.1: Let Ψ be an additive mapping on a ring R then Ψ is called generalized centralizerhomo on R into itself associated with centralizerhomo g on R is Ψ is called Jordan generalized centralizerhomo on R associated with Jordan centralizerhomo g on R if for all a∈ R ,then Ψ(ϰ 2)= g(ϰ) ϰ + Ψ(ϰ)g(ϰ) Ψ is called Jordan generalized triple centralizerhomo on R associated with Jordan triple centralizerhomo g on R if for all ϰ, u ∈ R ,then Ψ(ϰuϰ) = g(ϰ)uϰ + Ψ(ϰ)g(u)g(ϰ).
Lemma1 : Let R be a ring and Ψ is Jordan generalized centralizerhomo on R then, Comparing (1) and (2) we get Definition 2.3: Let be Jordan generalized centralizerhomo on a ring R then we define the mapping φ on In the following lemma we present the properties of φ: Holds for all ϰ, u ∈ R Proof: i) By lemma 1 we get iii) by using the same way of (ii) we get (iii).
Since R is 2 − torsion free we get Ψis generalized centralizerhomo.

Proposition 6:
Every Jordan generalized centralizerhomo of 2 − torsion free ring R is Jordan generalized triple centralizerhomo on R.
Proof: Let Ψbe Jordan generalized centralizerhomo on R.

Corollary 7:
Every generalized centralizerhomo on 2 − torsion free ring R is Jordan generalized triple centralizerhomo on R.
Proof: Let Ψ be generalized centralizerhomo on a ring R, then Ψ is Jordan generalized centralizerhomo on R, hence by Proposition 6 we obtain Ψ is Jordan generalized triple centralizedhomo on R.