Does there exist a characteristic $0$ principal ideal domain $R$ that has countably infinitely many prime ideals and such that there is no injective unital ring homomorphism $R\rightarrow \mathbb{C}$?
I am aware of examples of PIDs with countably many prime ideals coming from number theory but they are all subrings of $\overline{\mathbb{Q}}$. The PID $\mathbb{Q}[x]$ is not a subring of $\overline{\mathbb{Q}}$ but it is a subring of $\mathbb{C}$ via the homomorphism $x\rightarrow \pi$.
There exist fields of cardinality larger than continuum (at least assuming choice, not sure what happens otherwise) so that is a PID that does not embed into $\mathbb{C}$ but it does not have infinitely many prime ideals. On the other hand, the ring of univariate polynomials over such a field has more than countably many prime ideals.
Best Answer
Take an algebraically closed field $F$ of characteristic $0$ of cardinality larger than continuum (such a beast exists assuming choice, not sure otherwise). Consider the ring of univariate polynomials $F[x]$ and localize it at the multiplicative system of elements that are not multiples of $(x-n)$ for any positive integer $n$. The resulting PID can not embed in $\mathbb{C}$ because of cardinality and its prime ideals are $(0)$ and $(x-n)$ for positive integers $n$.