# Valuation Rings and Ultrafilters II

See my post here: Valuation Rings and Ultrafilters

Let $$K$$ be a field, and let $$\mathcal{S}$$ be the set of pairs $$(R, \mathfrak{p})$$ of subrings $$R$$ of $$K$$ with designated prime ideals $$\mathfrak{p}$$ of $$R$$. We order this set where $$(R , \mathfrak{p}) \leq (S, \mathfrak{q})$$ when $$R \subset S$$ and $$\mathfrak{q} \cap R = \mathfrak{p}$$. One of the elementary results about valuations is that valuations with fraction field $$K$$ are precisely the maximal elements of this set. This is sometimes even given as the definition.

The usual proof works by showing that we always have either $$(R, \mathfrak{p}) \leq (R[a], \mathfrak{p}R[a])$$ or $$(R, \mathfrak{p}) \leq (R[a^{-1}], \mathfrak{p}R[a^{-1}])$$ for each pair $$(R, \mathfrak{p})$$ in $$\mathcal{S}$$. Then, as valuations are those rings which always contain either $$a$$ or $$a^{-1}$$, the result follows.

I am trying to give an alternative proof of this fact, using a near correspondence between subrings of a ring $$R$$ and partially ordered abelian groups, but I can't seem to see how it works out, so I want some help.

See here for how, for a subring $$R$$ of a field $$K$$,

1) certain $$R$$ submodules of $$K$$ correspond to filters on $$\Lambda: = K^\times / R^\times$$

2) certain subrings of $$K$$ contianing $$R$$ correspond to filters on the partially ordered abelian group $$K^\times / R^\times$$ which are submonoids.

3) filters on $$K^\times / R^\times$$ which are submonoids correspond to extensions of the partial order on $$K^\times / R^\times$$.

This is a full correspondence when $$R$$ is a UFD, but for the general case, I think this could be close to an elegant proof that valuation rings are maximal in $$\mathcal{S}$$, just like the maximal extensions of the order on $$\Lambda$$ are total orders.

But things don't quite match up, since not all rings correspond to filters which are submonoids.

So, can anyone fix up this near-correspondence enough to get a clean proof that valuation rings are maximal elements of $$\mathcal{S}$$ in the general case?