Question 190
Let $m$ and $n$ be natural numbers $(n \geq 2)$. The ring $A$ has exactly $n$ elements and $a$ is an element of $A$ such that $1-a^{k}$ is invertible for every $k \in\{m+1, m+2, \ldots, m+n-1\}$. Prove that $a$ is nilpotent (that is, there exists a nonzero natural number $p$ such that $a^{p}=0$).