International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 625968, 14 pages
http://dx.doi.org/10.1155/2012/625968
Research Article

On Prime-Gamma-Near-Rings with Generalized Derivations

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

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

Received 26 January 2012; Accepted 19 March 2012

Academic Editor: B. N. Mandal

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 𝑁 be a 2-torsion free prime Γ -near-ring with center 𝑍 ( 𝑁 ) . Let ( 𝑓 , 𝑑 ) and ( 𝑔 , ℎ ) be two generalized derivations on 𝑁 . We prove the following results: (i) if 𝑓 ( [ 𝑥 , 𝑦 ] 𝛼 ) = 0 or 𝑓 ( [ 𝑥 , 𝑦 ] 𝛼 ) = ± [ 𝑥 , 𝑦 ] 𝛼 or 𝑓 2 ( 𝑥 ) ∈ 𝑍 ( 𝑁 ) for all 𝑥 , 𝑦 ∈ 𝑁 , 𝛼 ∈ Γ , then 𝑁 is a commutative Γ -ring. (ii) If 𝑎 ∈ 𝑁 and [ 𝑓 ( 𝑥 ) , 𝑎 ] 𝛼 = 0 for all 𝑥 ∈ 𝑁 , 𝛼 ∈ Γ , then 𝑑 ( 𝑎 ) ∈ 𝑍 ( 𝑁 ) . (iii) If ( 𝑓 𝑔 , 𝑑 ℎ ) acts as a generalized derivation on 𝑁 , then 𝑓 = 0 or 𝑔 = 0 .