Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 9E / b

The class $a$ is a unit modulo $p^e$ exactly when $\varphi (a)\ne 0$. This says a representative is not divisible by $p$, so it is coprime to $p^e$ and Bézout gives an inverse.

The class $a$ is nilpotent exactly when $\varphi (a)=0$. Divisibility by $p$ makes $a^e$ zero modulo $p^e$. Conversely, the image of a nilpotent in the field $R/(p)$ is nilpotent and therefore zero.