abstract algebra - When are u and 1-u never both units?

I was trying to generalize a solution to a problem (if $P\in \mathbb Z[x], n\in\mathbb Z,$ then $P(P(P(n)))\neq n$ unless $P(n)=n$), and I found a sufficient condition to replace $\mathbb Z$ with a domain $R$: $u$ and $1-u$ are never simultaneously units. Equivalently, $u(1-u)$ is never a unit in $R$ for $u\in R$. Unfortunately, the only examples I can find of rings with this property are the integers and the Gaussian integers.

Are there other rings with this property, or perhaps a more general condition that implies this property? What if we restrict ourselves to rings of integers of number fields?