This seems like an elementary question but a quick Google search did not yield an answer. I know that noetherian valuation rings are discrete and that discrete valuation rings have Krull dimension 1, but I'd like to know an example of a non-discrete (necessarily non-noetherian) valuation ring with Krull dimension 1. Or an explanation why such a ring cannot exist.
2026-03-26 13:51:43.1774533103
Non-discrete valuation rings of Krull dimension 1
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Exercise $2$ in Bourbaki, Commutative Algebra, ch. 6, Valuations, §3, explains the costruction of a valuation ring associated to a given totally ordered group $\Gamma$.
As height $1$ ordered groups are just subgroups of the ordered group $\mathbf R$, you just have to take a non-discrete subgroup of $\mathbf R$ for this construction, viz. $\mathbf Q$.