On gamma-rings with (σ, t )-skew-commuting and (σ, t )-skew-centralizing mappings

Abstract

Let M be a 2-torsion free Γ-ring with left identity e. Let D : M × M → M be a symmetric bi-additive mapping and d(x) = D(x, x). Let σ and τ be an endomorphism and an epimorphism of M , respectively. We prove the following:is a 3-torsion free Γ-ring satisfying xαyβz = xβyαz for all x, y, z ∈ M and α, β ∈ Γ, then 2-(σ, τ )-commutingness of d on M implies its (σ, τ )commutingness.