Co-commutators with generalized derivations in prime and semiprime rings

Abstract

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, F and G two nonzero generalized derivations of R, I an ideal of R and f(x1, . . . , xn) be a multilinear polynomial over C which is not central valued on R. If F (f(x1, . . . , xn))f(x1, . . . , xn)− f(x1, . . . , xn)G(f(x1, . . . , xn)) = 0 for all x1, . . . , xn ∈ I, then one of the following holds: (1) F (x) = xa and G(x) = xb for all x ∈ R with a = b ∈ C; (2) F (x) = xa and G(x) = bx for all x ∈ R with a = b; (3) F (x) = ax and G(x) = xb for all x ∈ R with a = b ∈ C; (4) F (x) = ax and G(x) = xb for all x ∈ R with a = b and f(x1, . . . , xn) is central valued on R; (5) F (x) = ax and G(x) = bx for all x ∈ R, with a = b ∈ C. We finally extend the result to a semiprime ring R in case F (x)x − xG(x) = 0 for all x ∈ R. Throughout this paperR always denotes an associative prime ring with center Z(R), extended centroid C and U its Utumi quotient ring. The Lie commutator of x and y is denoted by [x, y] and defined by [x, y] = xy − yx for x, y ∈ R. An Mathematics Subject Classification: 16N60, 16W25 .