Results on Multiplicative (Generalized) (α,β)-reverse Derivation on Prime Rings

Let ꭆ be a 2-torsion-free prime ring, U be non-zero square closed Lie ideal of ꭆ, α and β be automorphisms of ꭆ.A mapping Ϝ:ꭆ→ꭆ is called a multiplicative (generalized) (α,β)-reverse derivation if Ϝ(ab)=Ϝ(b)α(a)+β(b)d(a) for all a,b∈ꭆ where d: ꭆ → ꭆ is any map . The purpose of this paper, is to give some important results of multiplicative (generalized) (α,β)-reverse derivation on square closed Lie ideals Ϝ that satisfying any one of the properties: (i) Ϝ(uv)±α(uv)=0, (ii) Ϝ(uv)±α(vu)=0, (iii) Ϝ(u)Ϝ(v)±α(uv)=0, (iv) Ϝ(u)Ϝ(v)±α(vu)=0, (v) Ϝ(uv)=Ϝ(u)Ϝ(v), (vi) Ϝ(uv)=Ϝ(v)Ϝ(u) and (vii) Ϝ[u,v]=0 for all u,v∈U.