Some generalized identities on prime rings and their application for the solution of annihilating and centralizing problems

Abstract

Let R be a non commutative prime ring of characteristic different from 2, U be the Utumi quotient ring of R with the extended centroid C, f(x1, … , x_n) a multilinear polynomial over C which is not central valued on R, f(R) the set of all evaluations of the polynomial f (x₁, … , x_n). Suppose F and G are two nonzero generalized derivations on R and let p, q, ∈ R be such that f(R) satisfies the differential identity

In this paper we prove that, if R does not satisfy the standard identity s₄, then either R satisfies (1) or f(x₁, … , x_n)² is central valued on R. Moreover, in both cases we describe all possible forms of generalized derivations F and G.

Mathematics Subject Classification (2020):

• 16N60
• 16W25

Key words:

• Generalized derivations
• multilinear polynomials
• prime rings
• extended centroid
• Utumi quotient ring