Abstract

Let R be a ring and S a nonempty subset of R. Suppose that θ and ϕ are endomorphisms of R. An additive mapping δ:R→R is called a left (θ,ϕ)-derivation (resp., Jordan left (θ,ϕ)-derivation) on S if δ(xy)=θ(x)δ(y)+ϕ(y)δ(x) (resp., δ(x2)=θ(x)δ(x)+ϕ(x)δ(x)) holds for all x,y∈S. Suppose that J is a Jordan ideal and a subring of a 2-torsion-free prime ring R. In the present paper, it is shown that if θ is an automorphism of R such that δ(x2)=2θ(x)δ(x) holds for all x∈J, then either J⫅Z(R) or δ(J)=(0). Further, a study of left (θ,θ)-derivations of a prime ring R has been made which acts either as a homomorphism or as an antihomomorphism of the ring R.