Rings With $u^n-1$ Nilpotent For Each Unit $u$

We continue the study in-depth of the so-called $n$-UU rings for any $n\ge 1$, 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 $n\ge 2$, 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$\stackrel{\backslash c}{s}$an et al. in Hacettepe J. Math. Stat. (2020).