Left Annihilator of Identities with Generalized Derivations in Prime and Semiprime Rings

Let R be a noncommutative prime ring of char (R) ≠ 2, F a generalized derivation of R associated to the derivation d of R and I a nonzero ideal of R. Let S ⊆ R. The left annihilator of S in R is denoted by lR(S) and defined by lR (S) = {x ∈ R | xS = 0}. In the present paper, we study the left annihilator of the sets {F (x)◦n F (y)−x◦n y | x, y ∈ I} and {F (x)◦n F (y)−d(x◦n y) | x, y ∈ I}