TITLE: Example of a PID with a residue field of finite characteristic and a residue field of characteristic 0?
QUESTION [11 upvotes]: 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 (which proves much more), which I originally found at 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 2 3) on If a PID has no nonzero divisible elements, then is the same true of its finitely-generated modules?: $R = \mathbb Z_p[t]$ 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 $\mathbb Z_p[t]$.

REPLY [21 votes]: 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.