Geodesic
A note on periodic rings
Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 4, pp. 109-111 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {P. V. Danchev},
     title = {A note on periodic rings},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {109--111},
     year = {2021},
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     number = {4},
     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/

[1] Anderson D. D., Danchev P. V., “A Note on a Theorem of Jacobson Related to Periodic Rings”, Proceedings of the American Mathematical Society, 148:12 (2020), 5087–5089 | DOI | MR | Zbl

[2] Cui J., Danchev P., “Some New Characterizations of Periodic Rings”, Journal of Algebra and Its Applications, 19:12 (2020), 2050235 | DOI | MR | Zbl

[3] Abyzov A. N., Tapkin D. T., “On Rings with $x^n-x$ Nilpotent”, Journal of Algebra and Its Applications, 21 (2022) | DOI | MR

[4] Abyzov A. N., Danchev P. V., Tapkin D. T., “Rings with $x^n+x$ or $x^n-x$ Nilpotent”, Journal of Algebra and Its Applications, 22 (2023) | DOI | MR