I seem to have come up with a proof of the following statement, but I have not managed to find this statement, either as a proposition or as an exercise in some of the Abstract Algebra books I have looked at. So is this true (?):
Let $R$ be a UFD, $F$ its field of fractions, and $f \in R[x]$ a non-constant irreducible polynomial. If $g \in R[x]$ is a polynomial such that $f|g$ in $F[x]$, then $f|g$ in $R[x]$.