On Jordan ideals and left (<i>θ</i>, <i>θ</i>)‐derivations in prime rings

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 ( 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 .