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On torsion-free periodic rings. (English) Zbl 1090.16008
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 $$R^{n+1}=\{0\}$$.
##### MSC:
 16P99 Chain conditions, growth conditions, and other forms of finiteness for associative rings and algebras 16U99 Conditions on elements 20K15 Torsion-free groups, finite rank
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