On Prime Ring with Reverse (𝛂, 𝛃)-Derivations

Abstract

Throughout this research paper,R represent a prime ring. An additive mapping d∶ R→ R is said to be a derivation if it satisfies d(xy) = d(x)y + xd(y) for all x,y ∈R, alsod∶ R→ R is called a reverse derivation if it satisfies d(xy) = d(y)x + yd(x) for all x,y ∈R. A mapping F:R→R is said to be generalized derivation associated with derivation d∶ R→ R if F(xy)=F(x)y+xd(y) for all x,y ∈R and F:R→R is called a generalized reverse derivation associated with reverse derivation d:R→R if for all x,y∈R, thenF(xy)=F(y)x+yd(x).An additive mapping d: R →R associated with an automorphism α,β is called a reverse (α,β)-derivation on R if satisfying (xy)=d(y)α(x)+ β(y)d(x) , for all x,y∈R.The aim of this research paper is to establish the commutativity results of prime ring R by reverse (α,β)-derivation which satisfying certain constraints.