Question 184
Let $m$ and $n$ be positive integers, $n \ge 2$, let $R$ be an $n$-element ring, and let $x$ be an element of $R$ such that $1 - x^k$ is invertible for each $k \in \{m+1, m+2, \dots, m+n-1\}$. Show that $x$ is nilpotent (i.e., $x^p = 0$ for some positive integer $p$).