On Jordan ideals and left (θ, θ)‐derivations in prime rings

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.