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DOI: 10.46698/q0369-3594-2531-z

# 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