International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 646587, 6 pages
doi:10.1155/2010/646587
Research Article

Remarks on Generalized Derivations in Prime and Semiprime Rings

Basudeb Dhara

Department of Mathematics, Belda College, Belda, Paschim Medinipur 721424, India

Received 15 August 2010; Accepted 28 November 2010

Academic Editor: Hans Keiding

Copyright © 2010 Basudeb Dhara. 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 with center 𝑍 and 𝐼 a nonzero ideal of 𝑅 . An additive mapping 𝐹 ∶ 𝑅 → 𝑅 is called a generalized derivation of 𝑅 if there exists a derivation 𝑑 ∶ 𝑅 → 𝑅 such that 𝐹 ( 𝑥 𝑦 ) = 𝐹 ( 𝑥 ) 𝑦 + 𝑥 𝑑 ( 𝑦 ) for all 𝑥 , 𝑦 ∈ 𝑅 . In the present paper, we prove that if 𝐹 ( [ 𝑥 , 𝑦 ] ) = ± [ 𝑥 , 𝑦 ] for all 𝑥 , 𝑦 ∈ 𝐼 or 𝐹 ( 𝑥 ∘ 𝑦 ) = ± ( 𝑥 ∘ 𝑦 ) for all 𝑥 , 𝑦 ∈ 𝐼 , then the semiprime ring 𝑅 must contains a nonzero central ideal, provided 𝑑 ( 𝐼 ) ≠ 0 . In case 𝑅 is prime ring, 𝑅 must be commutative, provided 𝑑 ≠ 0 . The cases (i) 𝐹 ( [ 𝑥 , 𝑦 ] ) ± [ 𝑥 , 𝑦 ] ∈ 𝑍 and (ii) 𝐹 ( 𝑥 ∘ 𝑦 ) ± ( 𝑥 ∘ 𝑦 ) ∈ 𝑍 for all 𝑥 , 𝑦 ∈ 𝐼 are also studied.