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Existence of prime ideals and Axiom of Choice.,

I have found that The existence of prime ideals in commutative rings is equivalent to the Boolean Prime Ideal theorem. But $BPI$ is weaker than Axiom of choice. this means that The existence of prime ideal in commutative rings with unity is weaker than $AC$. Know Another Question came in my mind that I think It is a bit different from that one. Let me recall the following theorem:

**Theorem:For any commutative unitary ring $R$ there exists a minimal prime ideal.**

To proving this result One can pickup a prime ideal, and throw it in a maximal chain of prime ideals(Zorn's lemma) and then the intersection of this chain gives a minimal prime ideal at hand.

You Know that the existence of minimal prime ideal needs to apply one of the equivalences of $AC$ (i.e.Zorn's Lemma) But I didn't see anything about the converse of Above theorem.

**STATEMENT:Is it true that The existence of minimal prime ideals in commutative unitary rings is equivalent to $AC$**.

I am interested in To Know if the situation changes When we give minimality Condition on Prime ideals.

I think its better to recall the difference of two following situations in topology:

The statement "product of compact Hausdorff spaces is compact", does not implies $AC$

But

The statement "product of compact spaces is compact" is equivalent to $AC$

a lot morethan it is needed. People usually don't mind how much choiceisneeded, at best they would like to know if the axiom of choice is used, and rarely whether or not it has been used to its full extent. In this aspect for most people BPI is practically AC, and DC is practically AC - despite both being pretty far from actually being AC in its full glorious power. – Asaf Karagila Jun 3 '12 at 19:54