I encounter the following result in a lecture notes about algebraic curves:
Let $R,S$ be noetherian valuation rings over their common field of fractions $K$, and $R,S$ are not fields. Let $M=R\setminus R^*$, $N=S\setminus S^*$, then:
- $M,N$ are principal ideals;
- $R,S$ are maximal subrings of $K$;
- $M\subseteq N\Leftrightarrow M=N\Leftrightarrow R=S\Leftrightarrow R\subseteq S$.
For the 2nd statement, I am unsure how to prove. It is not hard to show that $R$, $S$ are maximal valuation rings of $K$, but I am not sure why $K$ cannot have any bigger subrings that does not admit a valuation.
Could anyone confirm me that the original statement is correct by giving a clue of proof, or give a counterexample that says it is not true?
Thank you very much!