Let $R$ be a commutative noetherian UFD (which is not a PID) of zero characteristic, $T$ an indeterminate, $h \in R[T]$, and denote by $h'$ the formal derivative of $h$.
Is there a criterion for primality of the ideal generated by $h$ and $h'$?
Of course, $(h,h')$ is a prime ideal iff $R[T]/(h,h')$ is a domain, but this is not a kind of answer I am looking for; it would be nice if there is an answer as follows: $(h,h')$ is a prime ideal iff "something about $R \to R[T]/(h)$; $R \to R[T]/(h')$; $R \to R[T]/(h,h')$; higher derivatives; conditions on $R$".
Remarks: (1) I do not think that assuming one of $h,h'$ is irreducible will help (but maybe I am wrong). (2) The "inspiration" for the question came from the following (true) claim: Either $h=0$ or $(h,h')=R[T]$ iff $R \to R[T]/(h)$ is formally smooth.
I hope my question is not nonsense or too general (I slightly suspect it may be too general; I hope it is possible to get a nice answer).