Let $R$ be an integral domain, let $\mathrm{Frac}(R)$ denote the field of fractions of $R$. Then as $\mathrm{Frac}(R)$ contains $R$ it is an $R$-module in an usual way. Now suppose every proper submodule of the $R$-module $\mathrm{Frac}(R)$ is cyclic. Then what can we say about $R$ ? Can such integral domains be characterized in some way ?
If I wanted the field of fractions $\mathrm{Frac}(R)$ itself to be cyclic (or even finitely generated) as an $R$-module then I know that $R$ would become a field. But my condition is only on proper submodules.