Exercise 11, § 1

(a) Let A be a Noetherian ring and E an A-module. Show that the canonical homomorphism of E to the product $\prod_{p \in \operatorname{Ass}(E)} E_p$ is injective (if N is the kernel of this homomorphism, show that $\operatorname{Ass}(N) = \varnothing$).

(b) Take A to be the polynomial ring $K[X, Y, Z]$ over a field K; let $p_1 = AX + AY, p_2 = AX + AZ$, which are prime ideals and a the ideal $p_1 p_2$. The set $\operatorname{Ass}(A/a)$ consists of $p_1, p_2$ and the maximal ideal $m = p_1 + p_2$; show that the canonical homomorphism from $E = A/a$ to $E_{p_1} \times E_{p_2}$ is not injective.