A Note on Periodic Rings

Danchev, P. V.
Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 4.

Abstract:
We obtain a new and non-trivial characterization of periodic rings (that are those rings \(R\) for which, for each element \(x\) in \(R\), there exists two different integers \(m\), \(n\) strictly greater than \(1\) with the property \(x^m=x^n\)) in terms of nilpotent elements which supplies recent results in this subject by Cui-Danchev published in (J. Algebra & Appl., 2020) and by Abyzov--Tapkin published in (J. Algebra & Appl., 2022). Concretely, we state and prove the slightly surprising fact that an arbitrary ring \(R\) is periodic if, and only if, for every element \(x\) from \(R\), there are integers \(m>1\) and \(n>1\) with \(m\not= n\) such that the difference \(x^m-x^n\) is a nilpotent.

Keywords: potent rings, periodic rings, nilpotent elements

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