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

  • $\begingroup$ Isn't this just a convoluted way of saying the maximal ideals are principal? Or maybe you are working without choice or identity and prefer not to phrase it that way? $\endgroup$ – rschwieb Aug 22 '16 at 0:03
  • $\begingroup$ @rschwieb : no no , I am okay with choice and my rings are commutative with unity ... but how do you arrive at your conclusion ? I am interested to see , please elaborate $\endgroup$ – user228168 Aug 22 '16 at 4:11
  • $\begingroup$ Your hypothesis must apply to each maximal ideal, and since the principal ideal containing the maximal ideal is proper, it must be equal to the maximal ideal. Thereafter, every proper ideal is contained in a maximal (and principal) ideal. $\endgroup$ – rschwieb Aug 22 '16 at 20:19

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