Pair of generalized derivations acting on multilinear polynomials in prime rings

Abstract

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C and f (r 1 , . . . , r n ) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two nonzero generalized derivations of R such that G ̸ = Id (identity map) andThen one of the following holds:(1) there exist λ ∈ C and µ ∈ C such that F (x) = λx and G(x) = µx for all x ∈ R with 2µ = 1;(2) there exist λ ∈ C and p, q ∈ U such that F (x) = λx and G(x) = px + xq for allx ∈ R with p + q ∈ C, 2(p + q) = 1 and f (x 1 , . . . , x n ) 2 is central valued on R;(3) there exist λ ∈ C and a ∈ U such that F (x) = [a, x] and G(x) = λx for all x ∈ R with f (x 1 , . . . , x n ) 2 is central valued on R; (4) there exist λ ∈ C and a, b ∈ U such that F (x) = ax + xb and G(x) = λx for allx ∈ R with a + b ∈ C, 2λ = 1 and f (x 1 , . . . , x n ) 2 is central valued on R; (5) there exist a, p ∈ U such that F (x) = xa and G(x) = px for all x ∈ R, with (p − 1)a = −ap ∈ C and f (x 1 , . . . , x n ) 2 is central valued on R; (6) there exist a, q ∈ U such that F (x) = ax and G(x) = xq for all x ∈ R with a(q − 1) = −qa ∈ C and f (x 1 , . . . , x n ) 2 is central valued on R.