In Atiyah and Macdonald, an exercise states:
Let A be a valuation ring of a field K. Show that every subring of K which contains A is a local ring of A.
I tried looking up what this means and found two (a priori) different characterizations. One says that $B$ is a local ring of $A$ is $B$ contains $A$ and their maximal ideals coincide. Another says that $B$ is isomorphic to $S^{-1}A$ for some multiplicatively closed set $S\subset A$.
I don't think it's the former, as if $A\subset B\subset K$ then $B$ is immediately a valuation ring and thus local. If we let $\mathfrak m_A$ and $\mathfrak m_B$ denote the maximal ideals of $A$ and $B$, and $x\in B - A$, then $x$ is a unit in $B$ as $x^{-1}\in A\subset B$. Moreover, $x^{-1}$ is a non-unit in $A$ (as $x\notin A$) so $x^{-1}\in\mathfrak m_A - \mathfrak m_B$. So it cannot be the first characterization as then the exercise would be false in the cases where $B\neq A$.
I would accept that it's the latter definition, but it's from a random untrustworthy source and so I fear that it may be incorrect and that instead there's a third spooky(!!) definition waiting to creep up on me.
Note: I don't need help with the exercise itself, just the definition. I have a proof (barring any potential mistakes that I need to check over) that works in the case where the definition is that $B\cong S^{-1}A$ for some $S$.
