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

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.