On Prime Near-Rings with Generalized Derivation

LetN be a zero-symmetric left near-ring, not necessarily with amultiplicative identity element; and letZ be its multiplicative center. DefineN to be 3-prime if for all a, b ∈ N\{0}, aNb / {0}; and callN 2-torsion-free if N, has no elements of order 2. A derivation onN is an additive endomorphism D of N such that D xy xD y D x y for all x, y ∈ N. A generalized derivation f with associated derivation D is an additive endomorphism f : N → N such that f xy f x y xD y for all x, y ∈ N. In the case of rings, generalized derivations have received significant attention in recent years. In 1 , we proved the following.