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 ( x y ) = θ ( x ) δ ( y ) + ϕ ( y ) δ ( x ) (resp., δ ( x 2 ) = 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 ( x 2 ) = 2 θ ( 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 .