Let $K$ be a field, and let $S$ be the set of local subrings of $K$. Put an order $\leq$ on $S$ where $A \leq B$ when $A \subset B$ and the maximal ideal of $A$ is sent into the maximal ideal of $B$. Then the valuation subrings of $K$ are the maximal elements with respect to this order.
Valuation rings don't have to be maximal under $\subset$ though. It seems like those are the rank-$1$ valuation rings, i.e. DVR's. This question is to confirm this. Actually here is what I want to ask:
Claim 1: Let $S$ be the set of valuation rings of $K$. Are the maximal such valuation rings discrete valuation rings?
and the related claim:
Claim 2: Let $S$ be the set of subrings of $K$ whose fraction field is $K$. Put an order $\leq$ on $S$ where $A \leq B$ when $A \subset B$. Then the discrete valuation subrings of $K$ are the maximal elements with respect to this order.
It seems like 1 is clear since if it is not a DVR (suppose it has value group $\Lambda$) then we can take a special $\lambda \in \Lambda$ with negative valuation and replace $A$ with $A[a]$ where $a \in K$ has value $\lambda$. Edit: actually this only works for valuation rings with finitely generated value group, if at all.
Could someone confirm whether this claim holds?