International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 37, Pages 1957-1964
doi:10.1155/S0161171204309075
On Jordan ideals and left (θ,θ)-derivations in prime rings
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
Received 8 September 2003
Copyright © 2004 S. M. A. Zaidi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.