Exercise 11, § 3

Let A be a ring, a an ideal of A and S the multiplicative subset of A consisting of the elements whose canonical images in $A/a$ are not divisors of 0; let $\Phi$ be the set of maximal elements of the set of ideals of A not meeting S (such that the ideals $p \in \Phi$ are prime, cf. § 2, no. 5, Proposition 11). Show that a is the intersection of its saturations with respect to the ideals $p \in \Phi$ (reduce it to the case where $a = 0$ and use Corollary 1 to Theorem 1).