Exercise 7, § 1

Let A be a Noetherian ring, M a finitely generated A-module and $a = \operatorname{Ann}(M)$.

(a) Show that, if a is prime, a is the least element of $\operatorname{Ass}(M)$. (Note that $a = \bigcap_{p \in \operatorname{Ass}(M)} p$.)

(b) Show that every prime ideal associated with $A/a$ is associated with M. (Note that if p is a prime ideal of A the annihilator of the class in $A/a$ of an element $a \in A$, then $p = \operatorname{Ann}(aM)$ and use (a).)

(c) Let p, q be two distinct prime ideals of A such that $p \subset q$. Show that, if $M = (A/p) \oplus (A/q)$, then $\operatorname{Ass}(A/a) \neq \operatorname{Ass}(M)$.