Generalized Derivations With Left Annihilator Conditions in Prime and Semiprime Rings

Abstract

Let R be a prime ring with its Utumi ring of quotients U, C = Z(U) be the extended centroid of R, H and G two generalized derivations of R, L a noncentral Lie ideal of R, I a nonzero ideal of R. The left annihilator of S ⊆ R is denoted by lR(S) and defined by lR(S) = {x ∈ R| xS = 0}. Suppose that S = {H(un)un +unG(un) | u ∈ L} and T = {H(xn)xn +xnG(xn) | x ∈ I}, where n ≥ 1 is a fixed integer. In the paper, we investigate the cases when the sets lR(S) and lR(T ) are nonzero.

Keywords

• prime ring
• derivation
• lie ideal
• generalized derivation
• ex- tended centroid
• utumi quotient ring
• 16w25
• 16w80
• 16n60