Exercise 21, § 2

Let $A$ be a commutative ring and $\mathscr{F}$ a topologizing and idempotent set of ideals of $A$. (a) Then the ring $A_{\mathscr{F}}$ is commutative (note that, if $m \in \mathscr{F}$ and

then $u(m^2) \subset m$; if $v$ is another element of $\operatorname{Hom}_A(m, A)$, show that for $x \in m, y \in m$. (b) Let $\mathscr{F}'$ be the family of ideals $l'$ of $A_{\mathscr{F}}$ such that $A_{\mathscr{F}}/l'$ is 9-negligible (exercise 19 (h)). Show that the mapping $p \mapsto p_{\mathscr{F}}$ is a bijection of the set of ideals $p$ of $A$ not belonging to $\mathscr{F}$ onto the set of prime ideals of $A_{\mathscr{F}}$ not belonging to $\mathscr{F}'$. (Show first that, if $A$ is an integral domain, so is $A_{\mathscr{F}}$, making the same note as in (a); then use the exact sequence of Exercise 19 (d) to show that $\mathfrak p_{\mathscr F}$ is prime. If $\mathfrak p'$ is a prime ideal of $A_{\mathscr F}$ not belonging to $\mathscr F'$ and $\mathfrak p=\bar j_A^{-1}(\mathfrak p')$, recall that $\mathfrak p'=\mathfrak p_{\mathscr F}$.)

(c) Let B be an A-algebra and $\mathscr G$ the set of left ideals I of B such that $B_s/I$ is $\mathscr F$-negligible. Show that $\mathscr G$ is the set of left ideals of B containing an ideal of the form $B\mathfrak m$, where $\mathfrak m\in\mathscr F$; deduce that $\mathscr G$ is topologizing and idempotent and that, for every B-module N, $\mathscr F N=\mathscr G N$ and $N_{\mathscr F}$ is canonically identified with $N_{\mathscr G}$.

(d) Let S be a multiplicative subset of A and $\mathscr F$ the set of ideals of A meeting S. Show that $\mathscr F$ is topologizing and idempotent and that, for every A-module M, $M_{(\mathscr F)}$ and $M_{\mathscr F}$ are canonically identified with $S^{-1}M$ (note that every ideal of $\mathscr F$ contains a principal ideal $As$, where $s\in S$).