Exercise 7, § 9

Let A be a commutative ring. (a) Every prime ideal is irreducible (§ 8, Exercise 11). (b) The set of prime ideals is inductive for the relations $\subset$ and $\supset$. (c) Let a be an ideal of A distinct from A; let b be the set of $x \in A$ such that there exists an integer $n \geq 0$ with $x^n \in a$ ($n$ depends on $x$). Show that b is an ideal and is equal to the intersection of the prime ideals of A containing a.