Given $R$ an absolute valued ring (with unit), sometimes $\overline{U(0,1)}=B(0,1)$ (for example, $\mathbb{Q},\mathbb{R},\mathbb{C},\mathbb{H}$) and sometimes $\overline{U(0,1)}\neq B(0,1)$ (for example, $\mathbb{Q}_p, \mathbb{Z}$).
Which properties must we impose to $R$ to ensure the identity holds or not?
Smells like the archimedean property has something to say here (p-adic example). But also $B(0,1)$ must be "full" in some sense (compare $\mathbb{Z}$ with $\mathbb{Z}[1/2]$, the latter satisfying the identity). Not sure if inverses really matter.
Also, all my counterexamples satisfy that the open ball is clopen. I wonder if there's a counterexample with a nonclosed open ball or if it is a necessary condition.
Edit: I forgot to say that $U(0,1)=\{x\in R\,:\,|x|<1\}$ and $B(0,1)=\{x\in R\,:\,|x|\leq 1\}$.