International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 37-40, Pages 1957-1964
doi:10.1155/S0161171204309075

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

S.M.A. Zaidi1 , Mohammad Ashraf2 and Shakir Ali1

1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
 2Department of Mathematics, Faculty of Science, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

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.