Let R be a prime ring with characteristic not 2 and σ,τ,λ,μ,α,β automorphisms of R. Let h be a nonzero left (resp. right)-generalized (σ,τ)-derivation of R and I,J nonzero ideals of R and a∈R. The main object in this article is to study the situations. (1) h(I)a⊂C_{λ,μ}(J) and ah(I)⊂C_{λ,μ}(J),(2) h(I)⊂C_{λ,μ}(J),(3) [h(I),a]_{λ,μ}=0, (4) h(I,a)_{λ,μ}=0 ( or (h(I),a)_{λ,μ}=0), (5) [h(x),x]_{λ,τ}=0,∀x∈I, (6) [h(x)a,x]_{λ,τ}=0,∀x∈I.