Given a commutative ring A. Prove that the set of prime ideals in A has minimal element with respect to inclusion.
I try to apply Zorn's lemma to prove above problem, but I can't find the way to construct a lower bound for any chain in prime ideals set. Please give me an idea to construct this. Thanks in advance!
Hint:
Consider the intersection of the prime ideals in a chain (the set of all prime ideals is not a chain, in general) and prove the contrapositive: if $x$ and $y$ do not belong to the intersection, neither does $xy$.