On Prime-Gamma-Near-Rings with Generalized Derivations

Abstract

Let 𝑁 be a 2-torsion free prime Γ-near-ring with center 𝑍(𝑁). Let (𝑓,𝑑) and (𝑔,ℎ) be two generalized derivations on 𝑁. We prove the following results: (i) if 𝑓([𝑥,𝑦]𝛼)=0 or 𝑓([𝑥,𝑦]𝛼)=±[𝑥,𝑦]𝛼 or 𝑓2(𝑥)∈𝑍(𝑁) for all 𝑥,𝑦∈𝑁, 𝛼∈Γ, then 𝑁 is a commutative Γ-ring. (ii) If 𝑎∈𝑁 and [𝑓(𝑥),𝑎]𝛼=0 for all 𝑥∈𝑁, 𝛼∈Γ, then 𝑑(𝑎)∈𝑍(𝑁). (iii) If (𝑓𝑔,𝑑ℎ) acts as a generalized derivation on 𝑁, then 𝑓=0 or 𝑔=0.