International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 270132, 14 pages
http://dx.doi.org/10.1155/2012/270132
Research Article

Generalized Derivations in Semiprime Gamma Rings

Kalyan Kumar Dey,1 Akhil Chandra Paul,1 and Isamiddin S. Rakhimov2

1Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh
2Department of Mathematics, FS, and Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 Serdang, Malaysia

Received 10 November 2011; Revised 5 December 2011; Accepted 5 December 2011

Academic Editor: Christian Corda

Copyright © 2012 Kalyan Kumar Dey 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 𝑀 be a 2-torsion-free semiprime Γ -ring satisfying the condition 𝑎 𝛼 𝑏 𝛽 𝑐 = 𝑎 𝛽 𝑏 𝛼 𝑐 for all 𝑎 , 𝑏 , 𝑐 ∈ 𝑀 , 𝛼 , 𝛽 ∈ Γ , and let 𝐷 ∶ 𝑀 → 𝑀 be an additive mapping such that 𝐷 ( 𝑥 𝛼 𝑥 ) = 𝐷 ( 𝑥 ) 𝛼 𝑥 + 𝑥 𝛼 𝑑 ( 𝑥 ) for all 𝑥 ∈ 𝑀 , 𝛼 ∈ Γ and for some derivation 𝑑 of 𝑀 . We prove that 𝐷 is a generalized derivation.