Exercise 5, § 1

(a) Let A be a Noetherian ring and $a$ an ideal of A. Show that there exists a finite number of prime ideals $p_i$ ($1 \leq i \leq r$) such that $p_1 p_2 \ldots p_r \subset a$. (Argue by reductio ad absurdum considering among the ideals containing a, distinct from A and containing no finite product of prime ideals, a maximal element;

observe on the other hand that, if the ideal b is not prime, there exist two ideals c, d containing b, distinct from b and satisfying cb ⊂ b.) (Cf. Chapter IV.)

If A is an integral domain and a ≠ 0, we may assume that the p_i ≠ 0.

(b) Give an example of a non-Noetherian ring A for which the result of (a) is false. (Take for example A to be the ring of continuous real-valued functions on [0, 1].)