Let $K$ be a field.
- A subring $R$ of $K$ is a valuation ring if for all nonzero $x\in K$, either $x$ or $x^{-1}$ belong to $R$.
- A valuation on $K$ is a map $v:K\to\mathbb{R}\cup\{\infty\}$ such that (i) $v(x)=\infty \iff x=0$ (ii) $v(xy)=v(x)+v(y)$ for all $x,y\in K$ (iii) $v(x+y)\ge \min(v(x),v(y))$ for all $x,y\in K$
It can be shown that if $v$ is a valuation on a field $K$, then $R := \{a\in K : v(a)\ge 0\}$ is a valuation ring.
Is it true that every valuation ring for the field $K$ can be written in the form above for some valuation $v$?
I did find a related post, but (i) the pdf referenced in the post is a broken link and (ii) it assumes that there is already an underlying valuation and the goal is to reconstruct it; my question is whether there always exists an underlying valuation. As such, I would appreciate if one does not simply refer me back to that post.
As mentionned in the comments, this is true if one allows $v$ to take values in any ordered group $\Gamma$, possibly non archimedean.
Given a valuation ring $R$, the group $\Gamma:= K^{\times}/ \ R^{\times}$ can be ordered by $aR^{\times} \leq bR^{\times}$ iff $bR \subseteq aR$ (or equivalently $aR^{\times} < bR^{\times}$ iff $\frac{b}{a} \in R \smallsetminus R^{\times}$). Then the quotient map $v: K^{\times} \rightarrow \Gamma$ satisfies the properties of a valuation and $R = \{0\} \cup \{a \in K^{\times} \ | \ v(a) \geq 0\}$.
Example of non archimedean $K^{\times} / R^{\times}$: consider $K:=\mathbb{R}(x,\exp(x))$, the field of fractions of the function ring generated by the identity, the exponentials, and the real constants. Then the subring $R:=O(1)$ of functions ultimately bounded by some constant is a valuation ring of $K$ and in the value group, $(xR^{\times})^n > \exp(x)R^{\times}$ for any natural number $n$. In fact, the value group here is isomorphic to $\mathbb{Z}^2$ with lexicographic order: send $f(x) = \frac{D(x)}{N(x)}$ to $(m_{N(x)} - m_{D(x)},n_{N(x)}-n_{D(x)})$ where $m_{P(x)}$ denotes the greatest integer $k$ such that $\exp(kx)$ appears in monomials of $P(x)$ and $n_{P(x)}$ is the corresponding exponent of $x$ in that monomial (so $P(x) = \Theta(\exp(m_{P(x)}x)x^{n_{P(x)}})$ at $+\infty$ in Landau notations).
In fact, any ordered group can appear as $K^{\times} / R^{\times}$ using similar constructions: take any field $k$, form the formal $k$-algebra $k[x^{\Gamma}]$ of polynomials in $x^{\gamma}, \gamma \in \Gamma$ (with $x^{\gamma}x^{\gamma'} = x^{\gamma+\gamma'}$) with coefficients in $k$, which is a domain since $k$ is. Take the field of fractions $k(x^{\Gamma})$, and as valuation, take $v P(x) = \min\{ \gamma \in \Gamma \ | \ x^{\gamma}$ appears as a monomial of $P(x)\}$ for $P \neq 0$ in $k[x^{\Gamma}]$, naturally extended to the fraction field. Then the value group is $\Gamma$.
See also fields of Hahn series $k((x^{\Gamma}))$, which are valued fields that are the "biggest" ones with given value group ($\Gamma$) and residue field ($k$), while the former construction $k(x^{\Gamma})$ rather builds the "smallest" ones (with some caveat).