Exercise 9, § 1

Let A be a Noetherian ring and a an ideal of A.

(a) For an ideal b of A to satisfy a: $b \neq a$, it is necessary and sufficient that b be contained in a prime ideal $p \in \operatorname{Ass}(A/a)$.

(b) Let A be the polynomial algebra $K[X, Y, Z]$ in 3 indeterminates over a field K. Let $n = AX, m = AX + AY + AZ, a = n \cap m^2$. Show that there is a prime ideal p containing a such that a: $p \neq a$ but which is not a prime ideal associated with A.