A note on periodic rings

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.