There’s a theorem saying that for local ring $A$, DVR $\Leftrightarrow$ every non-zero fractional ideal of $A$ is invertible. I am now in the $\Leftarrow$ direction. I managed in proving that the unique maximal ideal $m$ is principal and that $A$ is Noetherian. However, to deduce that $A$ is a DVR, we also need the Krull dimension of $A$ to be 1, i.e. every non-zero prime ideal of $A$ should be maximal, how can I prove it? Any help is appreciated!
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How did you get Noetherian and $m$ is principal? Noetherian local ring whose maximal ideal is principal easily implies that every principal ideal is a power of the maximal ideal which gives that every ideal is principal. – reuns Dec 31 '22 at 23:48
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@reuns Thank you very much. I think I get it now. – julian Jan 01 '23 at 00:48