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I mean that $B$ is a Dedekind domain with fraction field $L$, which is a finite separable extension of a field $K$ that is the fraction field of a PID $A$ such that $B$ is the integral closure of $A$ in $L$.

If the question in the title is true, one can give a new proof of the unique factorization property of Dedekind domains like the proof for number rings given in Hecke's classical book "Lectures on the Theory of Algebraic Numbers."

Edit: R. van Dobben de Bruyn has given an ingenious counterexample in the comment below.

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  • $\begingroup$ I made some edits to the title and to the body of the question. Let me know if the edits accurately reflect what you wanted to ask. $\endgroup$
    – KConrad
    Commented May 11, 2021 at 3:06
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    $\begingroup$ If I'm understanding the question correctly, a counterexample can be constructed as follows: start with a quadratic ring of integers $\mathcal O_K$ with nontrivial class group, take a principal prime ideal $\mathfrak p = (\alpha)$ that is not the only prime lying over $(p) = \mathfrak p \cap \mathbf Z$, and look at $\mathcal O_K[1/\alpha]$. It is not a PID itself, and the only PID below it is $\mathbf Z$, whose integral closure is $\mathcal O_K$ and not $\mathcal O_K[1/\alpha]$. Of course for your argument, it suffices that $B$ is a localisation of an integral closure of a PID. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 11, 2021 at 3:19
  • $\begingroup$ @KConrad Thanks; this is precisely I wanted to ask. $\endgroup$
    – J.Li
    Commented May 11, 2021 at 3:55
  • $\begingroup$ @R.vanDobbendeBruyn Many thanks for your ingenious counterexample. $\endgroup$
    – J.Li
    Commented May 11, 2021 at 3:57
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    $\begingroup$ Essentially the same counterexample was given by Claborn in 1965, which as far as I know is the oldest counterexample to this conjecture. However these examples are still localizations of such integral closures of PIDs. I wonder if there are examples of Dedekind domains which are not of this form either. $\endgroup$
    – Wojowu
    Commented Apr 24, 2022 at 22:07

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