International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 25-28, 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
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.