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

Abstract: 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 Ko\c{s}an et al. (Hacettepe J. Math. & Stat., 2020). Some other closely related results of this branch are also established.