uncountable principal ideal domain with few units

Question

Does there exist an uncountable principal ideal domain with only countably many units (or, even, with only finitely many units) ?
The answer would be yes, if only unique factorization domain was required, for instance the ring of polynomials with uncountably many indeterminates over a finite field.

Answer 1

As I just learned, Stack Exchange explicitly encourage users to answer their own questions. So, here is one to my above question:
Yes, there exists an uncountable PID with even one unit:
Start with any finite or countable field $K$. Let $D$ be a polynomial ring over $K$ with uncountably many indeterminates. Then $D$ is a unique factorization domain (but, of course, not a PID).
Now, the essential argument is Theorem 4.5 in this paper of Heinzer and Roitman. With the notation in this paper, $I_\omega(D)$ is a PID, which contains $D$ as a subring and has the same units as $D$, hence $K \setminus \{0\}$.