I was wondering if the following holds: if $A$ is a valuation ring with maximal ideal $m_A$ and $B$ is a ring extension of $A$ contained in $Frac(A)$ then the only maximal ideal of $B$ is still $m_A$.
I am convinced that this result is true. But then I can't understand a result stated in Atiyah and Macdonald that sais that being $S$ the set of all local subrings of $Frac(A)$ ordered by the relation of domination ($A$, $B$ local rings, $B$ dominates $A$ if $B \supseteq A$ and $m_B$=$m_A\cap A$) then $A\in S$ is maximal if and only if $A$ is a valuation ring of $Frac(A)$.
If the previous result is true I can't see why a valuation ring should be maximal in $S$ when it would be enough to take any ring extension of $A$ contained in $Frac(A)$.
Thanks for the attention.