Exercise 1, § 2

A multiplicative subset $S$ of a ring $A$ is called saturated if the relation $xy \in S$ implies $x \in S$ and $y \in S$.

(a) For a subset $S$ of $A$ to be multiplicative and saturated, it is necessary and sufficient that $A - S$ be a union of prime ideals of $A$.

(b) Let $S$ be a multiplicative subset of $A$ and $\tilde{S}$ the set of $x \in A$ for which there exists $y \in A$ such that $xy \in S$. Show that $\tilde{S}$ is the smallest saturated multiplicative subset containing $S$, $A - \tilde{S}$ is the union of the prime ideals of $A$ not meeting $S$ and, for every $A$-module $M$, the canonical mapping from $S^{-1}M$ to $\tilde{S}^{-1}M$ is bijective.

(c) Let $S$ and $T$ be two multiplicative subsets of $A$. Show that the two following properties are equivalent: $(\alpha) \ S \subset \tilde{T}$; $(\beta)$ for every $A$-module $M$ such that $S^{-1}M = 0, \ T^{-1}M = 0$. (To see that $(\beta)$ implies $(\alpha)$, consider a quotient $A$-module $A/A_S$, where $s \in S$.)