Derivations Acting as Homomorphisms and as Anti-homomorphisms in σ-Lie Ideals of σ-Prime Gamma Rings

Abstract

Let U be a non-zero σ-square closed Lie ideal of a 2-torsion free σ-prime Γ-ring M satisfying the condition aαbβc = aβbαc for all a, b, c ∈ M and α, β ∈ Γ, and let d be a derivation of M such that dσ = σd. We prove here that (i) if d acts as a homomorphism on U , then d = 0 or U ⊆ Z(M), where Z(M) is the centre of M ; and (ii) if d acts as an anti-homomorphism on U , then d = 0 or U ⊆ Z(M).