For an integral domain $R$ let $\mathrm{Frac}(R)$ denote its field of fractions. Then $R$ is embedded in $\mathrm{Frac}(R)$ and we can consider $\mathrm{Frac}(R)$ as an $R$-module.

Can we characterize all non-field integral domains $R$ such that every proper non-zero submodule of the $R$-module $\mathrm{Frac}(R)$ is projective ?

If $R$ satisfies my condition, then since every $R$-submodule of $K$ is projective hence so are in particular the fractional ideals of $R$, and hence every non-zero fractional ideal of $R$ is invertible, thus $R$ is a Dedekind domain. But $R=\mathbb Z$ is a Dedekind domain which does not satisfy my condition . So the family of integral domains, that satisfies the condition I stated , should be Dedekind domain + something more ; I can't figure out what this something more should be .