Generalized Strong Commutativity Preserving Centralizers of Semiprime Î“- Rings

Shahed Ali Hamil,Abdulrahman H Majeed

doi:10.24996/ijs.2019.60.10.16

Abstract

In this paper, we introduce the concept of generalized strong commutativity (Cocommutativity) preserving right centralizers on a subset of a Î“-ring. And we generalize some results of a classical ring to a gamma ring.

Highlights

Let M and Γ be additive abelian groups
Every ring M is a Γ-ring with M = Γ
In [9], Hoque and Paul proved that every Jordan centralizer of a 2.torsion free semiprime Γ–ring satisfying a certain assumption is a centralizer. They proved in [10], if T is an additive mapping on a 2.torsion free semiprime Γ–ring M with a certain assumption such that T(aαbβa) = aαT(b)βa, for all a, b ∈ M and α, β ∈ Γ, T is a centralizer

Summary

Introduction

Let M and Γ be additive abelian groups. If there exists a mapping of M × Γ × M → M, (a, α, b)→aαb which satisfies the conditions (i) aαb ∈ M (ii) (a + b)αc = aαc + bαc, a(α + β)c = aαc + aβc, aα(b+ c) = aαb + aαc (iii) (aαb)βc = aα(bβc) for all a, b, c ∈ M and α, β ∈ Γ, M is called a Γ-ring. An additive mapping T : M → M is a left(right) centralizer if T(aαb) = T(a)αb (T(aαb) = aαT(b)) holds for all a, b ∈ M and α ∈ Γ. An additive mapping T : M → M is Jordan left(right) centralizer if T(aαa) = T(a)αa(T(aαa) = aαT(a)) for all a ∈ M, and α ∈ Γ.

Results

Conclusion