Jordan Higher Triple Left Resp. Right Centralizers of Prime Γ-Rings

Afrah Mohammed Ibraheem,Salah Mehdi Salih

doi:10.24237/djps.17.03.210-561-icps

Abstract

hrough this paper we define the higher triple left resp. right centralizers of a Γ-ring Ɠ, and study some properties of Jordan higher triple left resp. right centralizers of Ɠ, addition to we prove that: every Jordan higher triple left resp- right centralizer of a Γ-ring Ɠ is higher triple left resp. right centralizer f Ɠ when Ɠ is a 2-torsion free prime gamma ring. Prime Γ-ring, Higher left centralizer, Higher triple left centralizer, Jordan higher triple left centralizer.

Highlights

In 1964 Nobusawa [1] presented the notion of gamma ring, and in 1966 Barnes [2] ǥeneralized the concept of gamma ring, J
Within this paper we present and researchch higher tiple left resp-right centralizer and Jordan higher triple left resp-right centralizer of a gamma ring Ɠ, and prove that every Jordan higher triple left resp. right centralizer of a 2-torsion free prime Γ-ring Ɠ is a hiǥher triple left resp. right centralizer of Ɠ
Example 1.2: Let Ɠ bѐ Γ-ring, t = nεN be a higher triple left centralizer of Ɠ, let S= {(a, b)|a, b ∈ Ɠ} and Γ = {(α, α)|α ∈ Γ} where the addition and multiplication defined on S by (a1, b1) + (a2, b2) = (a1 + a2, b1 + b2) and (a1, b1)(α, α)(a2, b2) = (a1αa2, b1αb2), ∀ a1, a2, b1, b2 ∈ Ɠ, α ∈ Γ Let T: Ś →Ś, T= (Ţn)nεN be a family of additive mappings on Ś defined by Tn (a, b) = ( tn (a),ŧn( b) ), ∀ (ɑ, b) ∈ S, Ţ is a higher triple left centralizer on Ś

Summary

Introduction

In 1964 Nobusawa [1] presented the notion of gamma ring, and in 1966 Barnes [2] ǥeneralized the concept of gamma ring, J. ‫ كما درسنا خواص تمركزات جوردان‬، Γ-‫خلال هذا البحث قمنا بتعريف التمركزات الثلاثية العليا اليسرى واليمنى للحلقات‬ ‫ بالاضافة الى اننا قمنا ببرهنة ان كل تمركز جوردان ثلاثي عالي يساري او يميني‬، Γ-‫الثلاثية العليا اليسرى واليمنى للحلقات‬ Right centralizer of a 2-torsion free prime Γ-ring Ɠ is a hiǥher triple left resp. Right centralizer os Ɠ if ∀ ɑ,b,c ∈Ɠ, α εΓ and k∈N tk(aαbβc) = ∑ki=1 ti(a)αti−1(b) βti−1(c),

Results

Conclusion