Domain $R$ s.t. for any proper ideal $I$ , $\mathcal F_I:=\{(x):x\in R , I \subseteq (x) \ne R\}$ is non-empty implies it contains a minimal element?

Question

How do we characterize those integral domains $R$ , which are not field , such that for any proper ideal $I$ of $R$ , the family $\mathcal F_I:=\{(x):x\in R , I \subseteq (x) \ne R\}$ is non-empty implies the family contains a minimal element ? We note that the condition holds for UFD s and the reason behind the interest of such a condition is that If $R$ is such a domain and if each family $\mathcal F_I$ is indeed non-empty then $R$ is a PID , so in particular , if in a UFD , every proper ideal is contained in a proper principal ideal then the domain is a PID . Any reference or link concerning this type of domains will be highly appreciated