On Some Generalized Identities with Derivations on Multilinear Polynomials

Abstract

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, Z(R) the center of R, f(x1,…,xn) a non-central multilinear polynomial over K, d and δ derivations of R, a and b fixed elements of R. Denote by f(R) the set of all evaluations of the polynomial f(x1,…,xn) in R. If a[d(u),u] + [δ (u),u]b = 0 for any u ∈ f(R), we prove that one of the following holds: (i) d = δ = 0; (ii) d = 0 and b = 0; (iii) δ = 0 and a = 0; (iv) a, b ∈ Z(R) and ad + bδ = 0. We also examine some consequences of this result related to generalized derivations and we prove that if d is a derivation of R and g a generalized derivation of R such that g([d(u),u]) = 0 for any u ∈ f(R), then either g = 0 or d = 0.