On $(σ, τ)$-derivations of Prime Near-Rings-II - Ashraf, Mohammad; Ali, Shakir (Sarajevo Journal of Mathematics, 2024)
Let $N$ be a left near-ring and let $σ, τ$ be automorphisms of $N$. An additive mapping $d : N \longrightarrow N$ is called a $(σ, τ)$-derivation on $N$ if $d(xy) = σ (x)d(y) + d(x)τ (y)$~for all $x,y \in N$. In this paper, we obtain Leibniz' formula for $(σ, τ)$-derivations on near-rings which facilitates the proof of the following result: Let $n ≥ 1$ be an integer, $N$ be $n$-torsion free, and $d$ a $(σ,τ)$-derivation on $N$ with $d^{n}(N)=\{0\}$. If both $σ $ and $τ$ commute with $d^{n}$ for all $n ≥ 1$, then $d(Z)=\{0\}$. Further, besides proving some more related results, we investigate commutativity of $N$ satisfying either of the properties: $d([x,y])= 0,$ ~or~ $d(xoy)= 0$, for all $ x,y \in N$. 2000 Mathematics Subject Classification. 16W25, 16Y30