Let $A$ be an (Noetherian) integral domain (of dimension one), $K$ its quotient field, $B$ a subring of $K$ such that $A\subseteq B \subseteq K$. Can we determine the dimension of $B$ in general? what about if we require $B$ is a valuation ring? Is there some reference about this?
Maybe this is too vague, I am more concerning the following case: When $A$ is local Noetherian, and $B$ is a valuation ring which dominates $A$, and $B$ is a subring of $K$, can we determine the dimension of $B$ in this setting?
Thanks.