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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.

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It is part of the Krull-Akizuki Theorem that $B$ is Noetherian of dimension at most one. Thus it has dimension one iff it is not a field. See for instance Matusmura's Commutative ring theory for a full statement and proof.