ORTHOGONAL GENERALIZED ( 𝜎 , 𝜏 ) (σ,τ)-DERIVATIONS IN SEMIPRIME Γ Γ-NEAR RINGS

Abstract

Consider a 2-torsion-free semiprime \(\Gamma\)-near ring \(N\). Assume that \(\sigma\) and \(\tau\) are automorphisms on \(N\). An additive map \(d_1: N \to N\) is called a \((\sigma, \tau)\)-derivation if it satisfies \[d_1(u \alpha v) = d_1(u) \alpha \sigma(v) + \tau(u) \alpha d_1(v) \]for all \(u, v \in N\) and \(\alpha \in \Gamma\). An additive map \(D_1: N \to N\) is termed a generalized \((\sigma, \tau)\)-derivation associated with the \((\sigma, \tau)\)-derivation \(d_1\) if \[D_1(u \alpha v) = D_1(u) \alpha \sigma(v) + \tau(u) \alpha d_1(v)\]for all \(u, v \in N\) and \(\alpha \in \Gamma\). Consider two generalized\hspace{0.1cm} \((\sigma, \tau)\)-derivations \(D_1\) and \(D_2\) on \(N\). This paper introduces the concept of the orthogonality of two generalized \((\sigma, \tau)\)-derivations \(D_1\) and \(D_2\) and presents several results regarding the orthogonality of generalized \((\sigma, \tau)\)-derivations and \((\sigma, \tau)\)-derivations in a \(\Gamma\)-near ring.