Let $A$ be a commutative ring. Let $f$ be any non-zero element of $A$. Suppose that $A/fA$ has a composition series as an $A$-module. Then we say $A$ is a weakly Artinian ring (this may not be a standard terminology).
Can we prove the following theorem without Axiom of Choice?
Theorem Let $A$ be a weakly Artinian integrally closed domain. Then the following assertions hold.
(1) Every ideal of $A$ is finitely generated.
(2) Every non-zero prime ideal is maximal.
(3) Every non-zero ideal of $A$ is invertible.
(4) Every non-zero ideal of $A$ has a unique factorization as a product of prime ideals.
EDIT May I ask the reason for the downvotes? Is this the reason?
EDIT What's wrong with trying to prove it without using AC? A proof without AC is constructive. When you are looking for a computer algorithm for solving a mathematical problem, this type of a proof may provide a hint. At least, you can be sure that there is a constructive proof.