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Does there exist a characteristic $0$ uncountable principal ideal domain $R$ that has countably infinitely many prime ideals, that is not a localization of a PID with uncountably many prime ideals at a multiplicative set?

This question without the second condition has been answered https://math.stackexchange.com/a/3327968/693936 but using localization seems to be a cheap trick to me.

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    These are getting awefully specific in order to get an example that just "feels right" to you. How will a person answering this question know that their example will be the right one for you? – Arthur Aug 19 '19 at 16:58
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    @Arthur the questions you have linked to are separate questions. If in an answer to this question you provide an example satisfying the specific conditions stated in the question, I will consider that to be a right answer for this question. Note that I have accepted the answer to the first question so I do not understand what is the issue (I am not allowed to accept the answer to the second question by software, I will once it allows me to). –  Aug 19 '19 at 17:00
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    Sure. But for a third attempt, I really think you ought to have spent some time making sure you yourself know what you're actually after, rather than just once more add a new requirement that barely disqualifies the previous answer. It makes it a lot easier for us to actually help you, which is what many of us are here for. Sure, you ticked my answer to your first question as accepted, but I didn't really feel helpful. Help us help you, and everyone ends up happier in the end. – Arthur Aug 19 '19 at 17:07
  • @Arthur I see. I guess I have a slightly different vision of this forum (namely that this is a question-answer database, not a help forum). I am sorry if it appears that I have not invested enough thought into my questions. Note that I did answer the second question myself so I did invest some effort from my side too. –  Aug 19 '19 at 17:09
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    You might be right. It is originally meant to be a question-answer database I think, but it has become mostly a help forum. I'm not even consistent about it myself. It just seemed strange to ask three so similar questions after one another. But I can't say it's wrong. – Arthur Aug 19 '19 at 21:52
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    this question was posted on MO (https://mathoverflow.net/q/338871/144105) –  Aug 21 '19 at 16:56

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