On the mapping \(xy\to(xy)^n\) in an associative ring. (Q1774758)

Summary: 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 \(R^2\), 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.