Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951.
Can we prove the following theorem without Axiom of Choice? If the answer is affirmative, by using this, we can get many examples of Dedekind domains without using Axiom of Choice. This is a related question.
Theorem Let $A$ be a commutative ring. Let $B$ be an integrally closed $A$-algebra. Suppose $B/fB$ has a composition series as an $A$-module for every non-zero element $f$ of $B$. Then the following assertions hold.
(1) Every ideal of $B$ is finitely generated.
(2) Every non-zero prime ideal of $B$ is maximal.
(3) Every non-zero ideal of $B$ is invertible.
(4) Every non-zero ideal of $B$ has a unique factorization as a product of prime ideals.