Let $R = \mathbb{Z}/p^n\mathbb{Z}$ where $p$ is a prime, and $n \ge 1.$ Let $\mathcal{U}(R)$ denote the units of $R.$ Is it possible to write any element $x \in R$ as $x = up^e$ where $u \in \mathcal{U}(R)$ and $0 \le e \le n$? Under what conditions does this factorization exist?
I am reading a paper, and it seems they assume they can always find this factorization.