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?
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 $f$.