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International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 1, Pages 205-206
http://dx.doi.org/10.1155/S0161171292000255

Remarks on derivations on semiprime rings

1Department of Mathematics, Faculty of Education, Umm Al-Qura University, Taif, Saudi Arabia
2Department of Mathematics, Brock University, Ontario, St. Catharines, Canada

Received 31 December 1990; Revised 10 May 1991

Copyright © 1992 Hindawi Publishing Corporation. 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

We prove that a semiprime ring R must be commutative if it admits a derivation d such that (i) xy+d(xy)=yx+d(yx) for all x, y in R, or (ii) xyd(xy)=yxd(yx) for all x, y in R. In the event that R is prime, (i) or (ii) need only be assumed for all x, y in some nonzero ideal of R.