Exercise 17, § 1

Let $A$ be a ring and $M$ an $A$-module. A prime ideal $p$ of $A$ is said to be weakly associated with $M$ if there exists $x \in M$ such that $p$ is a minimal element of the set of prime ideals containing $\operatorname{Ann}(x)$; we denote by $\operatorname{Ass}_f(M)$ the set of ideals weakly associated with $M$. Then $\operatorname{Ass}(M) \subset \operatorname{Ass}_f(M)$.

(a) Show that the relation $M \neq 0$ is equivalent to $\operatorname{Ass}_f(M) \neq \varnothing$.

(b) For $a \in A$ to be such that $a_M$ is injective, it is necessary and sufficient that $a$ belong to no element of $\operatorname{Ass}_f(M)$ (note that, if $a$ belongs to the radical of an ideal $\operatorname{Ann}(x)$ where $x \neq 0$ in $M$, there exists $y \neq 0$ in $M$ such that $a \in \operatorname{Ann}(y)$). To show that the condition is necessary, reduce it, by considering the ring $A/\operatorname{Ann}(x)$, to proving that in a ring $A$ every element belonging to a prime ideal $p$ which is minimal in the set of prime ideals of $A$ is necessarily a divisor of 0 (Chapter II, § 2, no. 6, Proposition 12).)

For $a \in A$ to be such that $a_M$ is almost nilpotent, it is necessary and sufficient that $a$ belongs to all the elements of $\operatorname{Ass}_f(M)$.

(c) If $N$ is a submodule of $M$, then

(Note that, if $p$ is a prime ideal, $a \notin p$, $x \in M$ such that $ax \in N$ and $\operatorname{Ann}(x) \subset p$, then $\operatorname{Ann}(ax) \subset p$.)

(d) Let $S$ be a multiplicative subset of $A$ and $\Phi$ the set of prime ideals of $A$ not meeting $S$. Show that $p \mapsto S^{-1}p$ is a bijection of $\operatorname{Ass}_f(M) \cap \Phi$ onto $\operatorname{Ass}_f(S^{-1}M)$. (Observe that the inverse image under the canonical mapping $A \to S^{-1}A$ of the annihilator of an element $x/s$, where $x \in M, s \in S$, is the saturation with respect to $S$ of $\operatorname{Ann}(x)$.)

(e) Let $S$ be a multiplicative subset of $A$; show that, if $N$ is the kernel of the canonical homomorphism $M \to S^{-1}M$, $\operatorname{Ass}_f(N)$ is the set of $p \in \operatorname{Ass}_f(M)$ which meet $S$ and $\operatorname{Ass}_f(M/N)$ the set of $p \in \operatorname{Ass}_f(M)$ which do not meet $S$. (To prove the latter point, consider a prime ideal $q$ which is minimal in the set of those containing $\operatorname{Ann}(\overline{y})$, where $\overline{y} \in M/N$; note that, if $y \in \overline{y}$ and $t \in q$, there exists $c \notin q$ and an integer $n > 0$ such that $ct^n y \in N$ (Chapter 11, § 2, no. 6, Proposition 12); deduce first that $q \cap S = \varnothing$. There exists $s \in S$ such that $sc t^n y = 0$; conclude that there can be no prime ideal $q' \neq q$ such that $\operatorname{Ann}(y) \subset q' \subset q$.)

(f) For a prime ideal of $A$ to belong to $\operatorname{Supp}(M)$, it is necessary and sufficient that it contain an element of $\operatorname{Ass}_f(M)$.

(g) Show that, if $A$ is Noetherian, $\operatorname{Ass}_f(M) = \operatorname{Ass}(M)$.

(h) If $A$ is an absolutely flat ring, then $\operatorname{Ass}_f(A) = \operatorname{Spec}(A)$.

(i) For every $A$-module $E$ the canonical homomorphism from $E$ to the product $\prod_{p \in \operatorname{Ass}_f(E)} E_p$ is injective; in other words, the intersection of the saturations of $\{0\}$ in $E$ with respect to the ideals $p \in \operatorname{Ass}_f(E)$ is reduced to 0. Generalize Exercise 12 similarly.

(j) Let M be a finitely generated A-module. Show that the minimal elements of the set of prime ideals containing $a = \operatorname{Ann}(M)$ belong to $\operatorname{Ass}_f(M)$ (if $(x_i)_{1 \leq i \leq n}$ is a system of generators of M, show that such an ideal contains one of the $\operatorname{Ann}(x_i)$). Deduce that $\operatorname{Ass}_f(A/a)$ is contained in $\operatorname{Ass}_f(M)$ (if a prime ideal $p$ contains the annihilator of the class mod. $a$ of an element $a \in A$, note that $p$ contains the annihilator of the submodule $\alpha M$ of M); show that $\operatorname{Ass}_f(A/a)$, $\operatorname{Ass}(M)$ and the set of prime ideals containing $a$ have the same minimal elements.