D=-1,-2,-3,-7 and-11 are the only five negative square free integers for which the corresponding quadratic Rings are Euclidean domains. Also D=-19,-41,-43 and-167 are the only four square free negative integers for which the corresponding quadratic rings are PIDs but not EDs. Are there any negative square free integers <-167 for which the corresponding quadratic rings are UFDs but not PIDs? Thank you in advance for your valuable time.
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Assuming that by "quadratic rings" you mean the integer ring of a quadratic extension of $\Bbb Q$.
The answer is then "no".
The integer ring of any number field is a Dedekind domain. It is a standard result that a ring is a PID if and only if it is a Dedekind domain and a UFD.
The proof can be found e.g. here.
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More generally PIDs are precisely the UFDs of dimension $\le 1,,$ see the dupe. – Bill Dubuque Mar 16 '21 at 01:10