Generalized derivations with nilpotent values in semiprime rings

Abstract

Let R be a (semi-) prime ring with extended centroid C, let f(X₁, … , X_k) be a multilinear polynomial over C in k noncommutative indeterminates which is not central-valued on R and let g be a generalized derivation of R. In this paper, we completely characterize the form of g and the structure of R such that (g(f(x₁, … , x_k))^m − γf(x₁, … , x_k)ⁿ)^s = 0 for all x₁, … , x_k ∈ R, where γ ∈ C and m, n, s are fixed positive integers. Our results naturally improve and generalize the theorems obtained by Huang and Davvaz in [Generalized derivations of rings and Banach algebras, Comm. Algebra (2013); 43, 1188–1194] and the theorems recently obtained by De Filippis et al. in [Generalized derivations with nilpotent, power-central and invertible values in prime and semiprime rings, Comm. Algebra (2019); 47, 3025–3039]. Moreover, we describe a revised version of the theorem obtained by Huang in [On generalized derivations of prime and semiprime rings, Taiwanese J. Math. (2012); 16, 771–776.]

Mathematics Subject Classification (2020):

• 16N60
• 16W25
• 16R50

Key words:

• (Semi-)Prime ring
• generalized derivation
• generalized polynomial identity (GPI)
• differential identity (DI)