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Q1. Does there exist an ID R in which every non zero prime ideal of type pR is maximal ideal but R is not PID?

Q2. Does there exist an ID R in which every non zero prime ideal is maximal ideal but R is not PID?

If R is a UFD, then there does not exist such examples. But is it true if R is ID or FD?

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  • $\begingroup$ The first question specifies principal prime ideals, right? Apparently you mean to specify nonzero prime ideals because an integral domain in which $\{0\}$ is maximal is a field. $\endgroup$ – rschwieb Sep 22 '15 at 14:48
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    $\begingroup$ right @rschwieb should I edit it? $\endgroup$ – Sushil Sep 22 '15 at 14:48
  • $\begingroup$ you definitely need to add "nonzero" before someone posts a trivial answer. You could add a bit more of your own work, if you have it handy, as well. $\endgroup$ – rschwieb Sep 22 '15 at 14:49
  • $\begingroup$ @rschwieb sorry had not realised it before. Thanks I have edited it $\endgroup$ – Sushil Sep 22 '15 at 15:24
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For question 2, any Dedekind domain satisfies these conditions: they're integral domains of Krull dimension $1$. For instance, any ring of algebraic integers is a Dedekind domain, and for such rings being a UFD is equivalent to being principal.

In the particular case of quadratic integers, there are only $9$ imaginary number fields such that their ring of quadratic integers is principal. For real number fields, it is not even known whether there is an infinity of them with a principal ring of integers (it is a conjecture by Gauß).

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