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Question: Is there a classification of (noetherian if needed) integral domains with finitely many units ? (of course we can exclude fields as trivial examples)

Probably there are many such domains that I can not think of but maybe there is an overview article of what is known.

If it helps one can also assume some additional things like characteristic 0.

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    $\begingroup$ My guess is that if you choose a random ideal $I\leq R=\mathbb{Z}[x_1,\dotsc,x_n]$ with not too many generators, then $R/I$ will usually be an integral domain with $(R/I)^\times=\{1,-1\}$. If so, then there is no hope of a classification. $\endgroup$
    – Neil Strickland
    Commented Mar 31, 2021 at 13:43
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    $\begingroup$ This could be interesting to you: arxiv.org/abs/1701.02341 $\endgroup$
    – R.P.
    Commented Mar 31, 2021 at 13:50
  • $\begingroup$ Note: if $R \rightarrow S$ has finite kernel and finite cokernel then the same is true for $R^* \rightarrow S^*$. $\endgroup$
    – David Lampert
    Commented Apr 1, 2021 at 18:16

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