$R$ is an integral domain then shows that an ideal $I$ of $R[X]$ is either principal or of the form $(f, r), r \in R$. The hint is to use the Gauss lemma.
I started by assuming that $I$ is not principal, then I want to consider $G= \{f \in I| f(0) = 0\}$, pick the element in it with minimal degree, I think that $G$ should be generated by this element. But the usual argument of division algorithm cannot go through here, because $R[X]$ is not Euclidean. How should I finish the proof?
$R$ should be assumed to be a PID.