  We continue the study in-depth of the so-called $n$-UU rings for any $n\geq
1$, that were defined by the first-named author in Toyama Math. J. (2017) as
those rings $R$ for which $u^n-1$ is always a nilpotent for every unit $u\in
R$. Specifically, for any $n\geq 2$, we prove that a ring is strongly
$n$-nil-clean if, and only if, it is simultaneously strongly $\pi$-regular and
an $(n-1)$-UU ring. This somewhat extends results due to Diesl in J. Algebra
(2013), Abyzov in Sib. Math. J. (2019) and Cui-Danchev in J. Algebra Appl.
(2020). Moreover, our results somewhat improves the ones obtained by
Ko$\c{s}$an et al. in Hacettepe J. Math. Stat. (2020).

Rings and Algebras