I understand that for any nonempty set $S$ of characteristics, there exists a PID $R$ such that the set of characteristics of residue fields of $R$ (i.e. quotients by of $R$ by maximal ideals -- I'm not including the residue field at the generic point. Thanks to Steven Landsburg for pointing out this terminological ambiguity in the comments below) is precisely $S$. I learned this from a paper of Heitmann, [PID’s with specified residue fields](https://www.doi.org/10.1215/S0012-7094-74-04162-3) (which proves much more), which I originally found at [Exotic principal ideal domains](https://mathoverflow.net/questions/56513/exotic-principal-ideal-domains).


**Question:** What is a "nice" example of a PID $R$ such that $R$ has a residue field of characteristic 0 and a residue field of finite characteristic?


By "nice", I'd ideally mean that $R$ is not just custom-built for the purpose of providing such an example, and might be a ring I'd meet on the street one day. Failing that, I'd settle for a streamlined description of such a ring $R$ (in order to understand Heitmann's example one must wade through several layers of extra generality related to his more ambitious aims).


If we only require $R$ to be Noetherian, then YCor gave a simple example in the comments ([1](https://mathoverflow.net/questions/373448/if-a-pid-has-no-nonzero-divisible-elements-then-is-the-same-true-of-its-finitel:#comment945165_373448) [2](https://mathoverflow.net/questions/373448/if-a-pid-has-no-nonzero-divisible-elements-then-is-the-same-true-of-its-finitel:#comment945166_373448) [3](https://mathoverflow.net/questions/373448/if-a-pid-has-no-nonzero-divisible-elements-then-is-the-same-true-of-its-finitel:#comment945227_373448)) on [If a PID has no nonzero divisible elements, then is the same true of its finitely-generated modules?](https://mathoverflow.net/questions/373448/if-a-pid-has-no-nonzero-divisible-elements-then-is-the-same-true-of-its-finitel): $R = \mathbb Z\_p[t]$ has residue fields $\mathbb F\_p$ and $\mathbb Q\_p$ (the latter obtained by modding out by $(1-pt)$). Similarly, $\mathbb Z\_{(p)}[t]$ has residue fields $\mathbb F\_p$ and $\mathbb Q$. It would be nice if there were an example of a PID with this property just as "nice" as $\mathbb Z\_p[t]$.



Example of a PID with a residue field of finite characteristic and a residue field of characteristic 0?
You can take the ring of fractions $\frac{a}{b}$ with $a,b \in \mathbb Z[x]$, where $b$ is nonzero mod $p$ and nonzero mod $px-1$.


Given any polynomial $a$, we can remove all factors of $p$ and remove all factors of $px-1$, obtaining a polynomial that is nonzero mod $p$ and nonzero mod $px-1$. So every polynomial is a power of $p^i (px-1)^j$ times a unit for natural numbers $i,j$.


Because the ideal generated by $p$ and $px-1$ contains $1$, the ideal generated by $p^{i\_1} (px-1)^{j\_1}$ and $p^{i\_2} (px-1)^{j\_2}$ is also generated by $p^{ \min(i\_1,i\_2)} (px-1)^{\min(j\_1,j\_2) } $. So every ideal is generated by a single element of the form $p^i (px-1)^j$.


There are two maximal ideals, $(p),$ and $(px-1)$, whose quotients $\mathbb F\_p(x)$ and $\mathbb Q$ have characteristics $p$ and $0$ respectively.


