Above, $U(S)$ refers to the units of $S$.
This problem stems from reading the paper "Translates of Polynomials", where a fact about a ring isomorphism between $S[1/a]$ and $S[1/b]$ is proved under the assumption that $\phi (U(S)) = U(S)$.
The paper then proceeds to prove a Lemma, Sublemma, and Corollary of the Lemma to show that if $k$ is a field and $S$ is any finitely generated extension of $k$ (eg. $k[x_1, ..., x_n]$), then $\phi (U(S)) = U(S)$ holds.
It seems reasonable that the assumption of $\phi (U(S)) = U(S)$ was included to prove the fact regarding $S[1/a]$ because there would be some example(s) where such a condition does not hold.
Given the Lemma's result, I realize I must be working with a UFD that is either an infinite extension of a field or a finite extension of a non-field domain to conjure such an example. I also realize that a unit of $S$ must map to some product of the irreducible components of $b$, but my repeated attempts to find such an example have failed. This is what brings me here.
I am hoping you can help. Whatever assistance can be offered is appreciated. In advance, thanks!