(untitled) (European Journal of Pure and Applied Mathematics, 2018)
An additive mapping F: R â†’ R is called a generalized derivation on R if there exists a derivation d: R â†’ R such that F(xy) = xF(y) + d(x)y holds for all x,y âˆˆ R. It is called a generalized (Î±,Î²)âˆ’derivation on R if there exists an (Î±,Î²)âˆ’derivation d: R â†’ R such that the equation F(xy) = F(x)Î±(y)+Î²(x)d(y) holds for all x,y âˆˆ R. In the present paper, we investigate commutativity of a prime ring R, which satisï¬es certain diï¬€erential identities on left ideals of R. Moreover some results on commutativity of rings with involutions that satisfy certain identities are proved.