Generalized Derivations in Semiprime Gamma Rings

Kalyan Kumar Dey,Akhil Chandra Paul,Isamiddin S Rakhimov

doi:10.1155/2012/270132

Kalyan Kumar Dey, Akhil Chandra Paul + Show 1 more

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https://doi.org/10.1155/2012/270132

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Abstract

LetMbe a 2-torsion-free semiprimeΓ-ring satisfying the conditionaαbβc=aβbαcfor alla,b,c∈M, α,β∈Γ, and letD:M→Mbe an additive mapping such thatD(xαx)=D(x)αx+xαd(x)for allx∈M, α∈Γand for some derivationdofM. We prove thatDis a generalized derivation.

Highlights

Hvala 1 first introduced the generalized derivations in rings and obtained some remarkable results in classical rings
The authors consider an additive mapping G : R → R of a ring R with the property G x2 G x x xD x for some derivation D of R. They prove that G is a Jordan generalized derivation
The author proved that if I is an ideal of a noncommutative prime ring R, a is a fixed element of R and F is an generalized derivation on R associated with a derivation d the condition F x, a 0 or F x, a 0 for all x ∈ I implies d x λ x, a

Summary

Introduction

Hvala 1 first introduced the generalized derivations in rings and obtained some remarkable results in classical rings. The authors consider an additive mapping G : R → R of a ring R with the property G x2 G x x xD x for some derivation D of R. They prove that G is a Jordan generalized derivation. The authors proved that if F is a commuting generalized derivation of a semiprime ring. Atteya 6 proved that if U is nonzero ideal of a semiprime ring R and R admits a generalized derivation D such that D xy − xy ∈ Z R R contains a nonzero central ideal. If there exists a derivation d of M such that D xαx D x αx xαd x for all x ∈ M, α ∈ Γ, D is a Jordan generalized derivation

Preliminaries

Main Results

D A D xαyβzδyαx yαxβzδxαy