Table of Contents
ISRN Algebra
Volume 2011 (2011), Article ID 750382, 5 pages
http://dx.doi.org/10.5402/2011/750382
Research Article

Generalized Derivations and Left Ideals in Prime and Semiprime Rings

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

Received 19 May 2011; Accepted 7 July 2011

Academic Editors: V. De Filippis and A. Rapinchuk

Copyright © 2011 Basudeb Dhara and Atanu Pattanayak. 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 an associative ring, πœ† a nonzero left ideal of 𝑅, π‘‘βˆΆπ‘…β†’π‘… a derivation and πΊβˆΆπ‘…β†’π‘… a generalized derivation. In this paper, we study the following situations in prime and semiprime rings: (1) 𝐺(π‘₯βˆ˜π‘¦)=π‘Ž(xyΒ±yx); (2) 𝐺[π‘₯,𝑦]=π‘Ž(xyΒ±yx); (3) 𝑑(π‘₯)βˆ˜π‘‘(𝑦)=π‘Ž(xyΒ±yx); for all π‘₯,π‘¦βˆˆπœ† and π‘Žβˆˆ{0,1,βˆ’1}.