Geodesic
A note on periodic rings
Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 4, pp. 109-111

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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. 
@article{VMJ_2021_23_4_a11,
     author = {P. V. Danchev},
     title = {A note on periodic rings},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {109--111},
     publisher = {mathdoc},
     volume = {23},
     number = {4},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2021_23_4_a11/}
}
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P. V. Danchev. A note on periodic rings. Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 4, pp. 109-111. http://geodesic.mathdoc.fr/item/VMJ_2021_23_4_a11/