Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951.
We use the definitions in my answers to this question. Can we prove the following theorem without Axiom of Choice?
Theorem Let $A$ be a weakly Artinian integral 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 $A$. Then the following assertions hold.
(1) Every ideal of $B$ is finitely generated
(2) Every non-zero prime ideal of $B$ is maximal.
(3) $leng_A B/I$ is finite for every non-zero ideal $I$ of $B$.