One of the versions of the classical Gauss' lemma in abstract algebra states the following:
Theorem: Let $R$ be an integral domain, $f\in R[X]$ of positive degree and $K$ the quotient field of $R$. If $R$ is a UFD, these facts are equivalent:
i) $f$ is irreducible in $R[X]$.
ii) $f$ is primitive in $R[X]$ and irreducible in $K[X]$.
This result, as Wikipedia points out, it's still valid when $R$ is a GCD domain. So my question is basically this: what is the weakest hypothesis we can assume in $R$ and still be able to guarantee that the above result is true?
Certainly we would have to change the usual definition of "primitive polynomial". I think it's appropriate to take a definition that is used in many research papers, namely we say that $f=a_0+a_1X+\ldots +a_nX^n$ is primitive if $(a_0,a_1,\ldots a_n)\subseteq (d)$ implies $d\in R^{\times}$, i.e., the only elements that divide all the coefficients of $f$ are the units of $R$.
It's easy to see that this definition agrees with the standard one: if $R$ is a GCD domain then $f$ is primitive iff $\gcd(a_0,a_1,\ldots ,a_n)=1$.
Now, using the above definition it's easy to show that $f$ irreducible in $R[X]$ implies that $f$ is primitive in $R[X]$ and moreover that ii) implies i), without any extra assumption on $R$, but in general $f$ irreducible in $R[X]$ doesn't imply $f$ irreducible in $K[X]$.
This leads to the following question: what extra conditions we need to guarantee that $f$ irreducible in $R[X]$ implies that $f$ is irreducible in $K[X]$?
Is there any research papers about the question above?
