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, let F, G and H be three generalized derivations of R, I an ideal of R and f(x₁,..., x _n ) a multilinear polynomial over C which is not central valued on R. If

$$F(f(r))G(f(r)) = H(f(r)^2 )$$

for all r = (r₁,..., r _n ) ∈ Iⁿ, then one of the following conditions holds:

• (1) there exist a ∈ C and b ∈ U such that F(x) = ax, G(x) = xb and H(x) = xab for all x ∈ R

• (2) there exist a, b ∈ U such that F(x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C

• (3) there exist b ∈ C and a ∈ U such that F(x) = ax, G(x) = bx and H(x) = abx for all x ∈ R

• (4) f(x₁,..., x _n )² is central valued on R and one of the following conditions holds

  • (a) there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all x ∈ R, with ab = p + p’

  • (b) there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all x ∈ R, with p + p’ = ab ∈ C.