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Can we prove the following theorem without Axiom of Choice? This is a generalization of this problem.

Theorem Let $A$ be a weakly Artinian domain. Let $K$ be the field of fractions of $A$. Let $L$ be a finite extension field of $K$. Let $B$ be a subring of $L$ containing $A$. Let $P$ be a prime ideal of $B$. Then there exists a valuation ring of $L$ dominating $B_P$.

As for why I think this question is interesting, please see(particularly Pete Clark's answer): Why worry about the axiom of choice?

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