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I am wondering whether there are reasonable necessary and/or sufficient conditions to dedice whether a commutative ring can be uniquely ordered (like for instance $\mathbb{Z}$) or not. In the field ...

Let K be a field. Then a subring R of K is called a valuation ring if for all $x \in K^*,$ either $x \in R$ or $x^{-1} \in R$ (or both). It can be shown that for any valuation $v$ on $K,$ the ring $\...