Let M be a 2-torsion-free semiprime Γ-ring satisfying the condition aαbβc aβbαc for all a, b, c ∈ M, α, β ∈ Γ, and let D : M → M be an additive mapping such that D xαx D x αx xαd x for all x ∈ M, α ∈ Γ and for some derivation d of M. We prove that D is a generalized derivation.
Let N be a 2 torsion free prime Γ-near-ring with center Z(N ) and let d be a nontrivial derivation on N such that d(N ) ⊆ Z(N ). Then we prove that N is commutative. Also we prove that if d be a nonzero (σ,τ )-derivation on N such that d(N ) commutes with an element aofN then ether d is trivial or a is in Z(N ). Finally if d 1 be a nonzero (σ,τ )-derivation and d 2 be a nonzero derivation on N such thatthen N is a commutative Γ-ring.
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