Let $R$ be a noetherian integral domain, and $I$ a non-zero ideal of $R$ which can be generated by two elements. (We do not know if $I$, considered as an $R$-module, is $R$-projective; maybe yes maybe no).
My question: Can one somehow tell when such $I$ has a finite projective resolution?
Of course, if $I$ is $R$-flat, then it is $R$-projective (remember $R$ is noetherian), so it has a projective resolution of length $0$.
EDIT: (1) Demanding that $R$ has a finite global dimension is NOT an answer I am looking for.
(2) Demanding that $R$ is a UFD is NOT an answer I am looking for; please see The G-function of Macrae, Theorem 4.10 and Corollary 4.11. (4.10 says: $R$ is a UFD iff every two-generated ideal has a finite free resolution. 4.11 says: If every two-generated ideal has a finite projective resolution, then $R$ is integrally closed. I adjusted 4.10 to the noetherian case).