International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 26, Pages 1393-1396
doi:10.1155/S0161171204208250

On the mapping xy→(xy)n in an associative ring

Scott J. Beslin1 and Awad Iskander2

1Department of Mathematics and Computer Science, Nicholls State University, Thibodaux 70310, LA, USA
2Department of Mathematics, University of Louisiana at Lafayette, Lafayette 70504, LA, USA

Received 26 August 2002

Copyright © 2004 Scott J. Beslin and Awad Iskander. 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 consider the following condition (*) on an associative ring R:(*). There exists a function f from R into R such that f is a group homomorphism of (R,+), f is injective on R2, and f(xy)=(xy)n(x,y) for some positive integer n(x,y)>1. Commutativity and structure are established for Artinian rings R satisfying (*), and a counterexample is given for non-Artinian rings. The results generalize commutativity theorems found elsewhere. The case n(x,y)=2 is examined in detail.