Prime ideal being maximal ideal and PID

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?

• 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. – rschwieb Sep 22 '15 at 14:48
• right @rschwieb should I edit it? – Sushil Sep 22 '15 at 14:48
• 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. – rschwieb Sep 22 '15 at 14:49
• @rschwieb sorry had not realised it before. Thanks I have edited it – Sushil Sep 22 '15 at 15:24

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).