In every ring with unity satisfying ACC, every ideal is finitely generated; can we prove it without assuming Axiom of choice?

Question

Assuming Zorn's lemma, we can prove that in every ring with unity satisfying Ascending-Chain-Condition, every ideal is finitely generated. Is this statement equivalent to Zorn's lemma? Can we prove it without assuming Axiom of choice?

Answer 1

If you look at the fourth section of Hodges paper,

Wilfrid Hodges, Six impossible rings, J. Algebra 31 (1974), 218--244.

He proves (Section 3, Th. 1) that the implication that you are asking for is not provable without the axiom of choice. Namely, the following chain of implications is irreversible without using Zorn's lemma:

$$\text{Every non-empty set of ideals has a maximal element}\implies\text{Every ideal is finitely generated}\implies\text{Every strictly increasing chain of ideals is finite}$$

Of course, one can note by looking at the usual proof that with the principle of dependent choice, we can prove these are indeed equivalent. So it is strictly weaker than the axiom of choice.