Rings such that, for each unit \(u\), \(u^n-1\) belongs to the \(\Delta(R)\)

We study in-depth those rings \(R\) for which, there exists a fixed \(n\geq 1\), such that \(u^n-1\) lies in the subring \(\Delta(R)\) of \(R\) for every unit \(u\in R\). We succeeded to describe for any \(n\geq 1\) all reduced \(\pi\)-regular \((2n-1)\)-\(\Delta\)U rings by showing that they satisfy the equation \(x^{2n}=x\) as well as to prove that the property of being exchange and clean are tantamount in the class of \((2n-1)\)-\(\Delta\)U rings. These achievements considerably extend results established by Danchev (Rend. Sem. Mat. Univ. Pol. Torino, 2019) and Koşan et al. (Hacettepe J. Math. \& Stat., 2020). Some other closely related results of this branch are also established.