On torsion-free periodic rings. (Q2576029)

A ring \(R\) is called periodic if for each \(x\in R\), there exist distinct positive integers \(m,n\) such that \(x^n=x^m\). The principal theorem of this paper characterizes the rank-two torsion-free Abelian groups which admit a nontrivial periodic ring structure. Another result states that if \(R\) is a periodic torsion-free ring of rank \(n\), then \(R^{n+1}=\{0\}\).