If $U$ is a UFD, then there is a PID $P$ containing $U$ that has the same unit group as $U$. Moreover, $P$ can be taken so that its field of fractions is a pure transcendental extension of the field of fractions of $U$ with trans-cendence degree at most $|U|$.
"Extending UFDs to PIDs without adding units." J. Commut. Algebra 10 (3) 347 - 358, 2018. https://doi.org/10.1216/JCA-2018-10-3-347