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I already know that if $R$ is a Noetherian local ring with Krull dimension $1$, then $R$ is DVR if and only if its maximal ideal $\mathfrak{m}$ is principal ideal if and only if every nonzero ideal is a power of $\mathfrak{m}$.

From those facts, I think I can say that $R$ is PID. Is this true?

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  • $\begingroup$ Compare with the equivalent statements here. The proofs are just a few lines. So you only need $(1)\Rightarrow (2)$ to see that a DVR is Euclidean and hence a PID. $\endgroup$
    – Dietrich Burde
    May 16, 2021 at 8:19
  • $\begingroup$ @DietrichBurde Thank you for your comment. But I think the statement also followed by what I wrote in the post. I wonder if my thought is correct $\endgroup$
    – one potato two potato
    May 16, 2021 at 8:28
  • $\begingroup$ Yes, your thought is $(10)$ implying $(3)$ in the duplicate. This is also correct. $\endgroup$
    – Dietrich Burde
    May 16, 2021 at 8:30
  • $\begingroup$ It is obvious that in a DVR with $\pi$ of minimal non-zero valuation then the non-zero ideals are exactly the $(\pi^n)$ $\endgroup$
    – reuns
    May 16, 2021 at 12:38

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