μ-Centralizers with Commutativity Results of Prime and Semiprime Γ-Rings

Let A be a 2- torsion free prime Γ- ring, and L be a square closed Lie ideal of A i.e. xβx∈L for all x∈L and β∈Γ, suppose that μ is an endomorphism of A, and f is a right μ- centralizer which is not the identity map on L, satisfying f([x, b]_γ )=〖∓[x,b]〗_γ ( or f(xδr)=∓xδr) for all x,b∈L , γ∈Γ, then L⊆Z(A).