On Prime-Gamma-Near-Rings with Generalized Derivations

Let ð ‘ be a 2-torsion free prime Γ -near-ring with center ð ‘ ( ð ‘ ) . Let ( ð ‘“ , ð ‘‘ ) and ( ð ‘” , â„Ž ) be two generalized derivations on ð ‘ . We prove the following results: (i) if ð ‘“ ( [ ð ‘¥ , 𠑦 ] ð ›¼ ) = 0 or ð ‘“ ( [ ð ‘¥ , 𠑦 ] ð ›¼ ) = ± [ ð ‘¥ , 𠑦 ] ð ›¼ or ð ‘“ 2 ( ð ‘¥ ) ∈ ð ‘ ( ð ‘ ) for all ð ‘¥ , 𠑦 ∈ ð ‘ , ð ›¼ ∈ Γ , then ð ‘ is a commutative Γ -ring. (ii) If ð ‘Ž ∈ ð ‘ and [ ð ‘“ ( ð ‘¥ ) , ð ‘Ž ] ð ›¼ = 0 for all ð ‘¥ ∈ ð ‘ , ð ›¼ ∈ Γ , then ð ‘‘ ( ð ‘Ž ) ∈ ð ‘ ( ð ‘ ) . (iii) If ( ð ‘“ ð ‘” , ð ‘‘ â„Ž ) acts as a generalized derivation on ð ‘ , then ð ‘“ = 0 or ð ‘” = 0 .