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

Zahraa S. M.  Alhaidary; Abdulrahman H.  Majeed

doi:10.24996/ijs.2024.65.12.%g

Authors

Zahraa S. M. Alhaidary Branch of Mathematics and computer Applications, Department of Applied Sciences ,University of Technology, Baghdad- Iraq
Abdulrahman H. Majeed Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq

DOI:

https://doi.org/10.24996/ijs.2024.65.12.%25g

Keywords:

Prime Ring, Multiplicative (Generalized) (α , β) Reverse Derivation, Lie ideal

Abstract

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.