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It's a well-know theorem in (commutative) ring theory that if $I$ is an ideal of a commutative ring $R$, then its radical $\sqrt{I}$ is equal to the intersection of all the prime ideals of $R$ containing $I$. I've been searching for different proofs about this result and every proof that I've found uses Zorn's lemma, so my questions are: is there a proof of the result above without using Zorn's lemma? is this result equivalent to Zorn's lemma?

Finally, an historical question: who proved this theorem first?

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The statement that $\sqrt{I}$ is an intersection of primes implies the existence of prime ideals in nonzero commutative unital rings. In fact, they are equivalent: First, replace $R$ with $R/I$. If $f$ is not nilpotent, then pulling back a prime of $R_f$ gives a prime of $R$ not containing $f$.

What is the relationship between the existence of prime ideals and the axiom of choice? There is a lot of discussion of this in the mathoverflow thread here.

In short, the axiom of choice is equivalent to the existence of maximal ideals, but the existence of prime ideals is weaker, and equivalent to the Boolean prime ideal theorem, which is a widely-used weakening of the axiom of choice.

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