Generalized derivations with Engel condition on Lie ideals of prime rings

Abstract

Consider ℜ {\Re} as a prime ring which is non-commutative in structure with a suitable characteristic. Here, 𝒵 ⁢ ( ℜ ) {\mathcal{Z}(\Re)} is the center of ℜ {\Re} and 𝒬 {\mathcal{Q}} is the Utumi ring of quotients where 𝒞 {\mathcal{C}} is the extended centroid of ℜ {\Re} . Suppose 𝒫 {\mathcal{P}} to be a Lie ideal of ℜ {\Re} which is non-central. Let 𝒦 {\mathcal{K}} be a generalized derivation of ℜ {\Re} related with derivation μ of ℜ {\Re} . If 𝒦 {\mathcal{K}} satisfies certain typical algebraic identities, then we prove that 𝒦 {\mathcal{K}} is either the identity map or the zero map or the scalar map and further information is also drawn on the associated scalar unless ℜ {\Re} embeds in M 2 ⁢ ( 𝒞 ) {M_{2}(\mathcal{C})} , a matrix ring of order 2 × 2 {2\times 2} over 𝒞 {\mathcal{C}} .