Strong commutativity preserving derivations on Lie ideals of prime Γ-rings

Abstract

Let M be a Γ-ring and S ⊆ M. A mapping f : M → M is called strong commutativity preserving on S if [f(x), f(y)]α = [x, y]α, for all x, y ∈ S, α ∈ Γ. In the present paper, we investigate the commutativity of the prime Γ-ring M of characteristic not 2 with center Z(M) 6= (0) admitting a derivation which is strong commutativity preserving on a nonzero square closed Lie ideal U of M. Moreover, we also obtain a related result when a mapping d is assumed to be a derivation on U satisfying the condition d(u) ◦α d(v) = u ◦α v, for all u, v ∈ U, α ∈ Γ.

Keywords

• Prime gamma rings
• Lie ideals
• derivations
• strong commutativity preserving maps