Commutativity of Prime Γ-near Rings with Γ− (σ, Τ)-derivation
نویسندگان
چکیده
Let N be a prime Γ-near ring with multiplicative center Z. Let σ and τ be automorphisms of N and δ be a Γ− (σ, τ)-derivation of N such that N is 2-torsion free. In this paper the following results are proved: (1) If σγδ = δγσ and τγδ = δγτ and δ(N) ⊆ Z, or [δ(x), δ(y)]γ = 0, for all x, y ∈ N and γ ∈ Γ, then N is a commutative ring. (2) If δ1 is a Γ-derivation, δ2 is a Γ − (σ, τ) derivation of N such that τγδ1 = δ1γτ and τγδ2 = δ2γτ , then δ1(δ2(N)) = 0 implies δ1 = 0 or δ2 = 0. (3) The condition for a Γ − (σ, τ)-derivation to be zero in prime Γ-near ring is also investigated.
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