On Prime-Gamma-Near-Rings with Generalized Derivations

Abstract EN

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.