Generalized derivations with nilpotent values in semiprime rings

Let R be a (semi -)prime ring with extended centroid C, let f (X-1, ..., 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 (x(1), ..., x(k)))(m) - eta f (x(1), ..., x(k))(n))(s) = 0 for all x(1), ..., x(k) is an element of R, where gamma is an element of 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, powercentral 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.]