Exercise 2, § 2

Let $A$ be a ring and $S$ a multiplicative subset of $A$. Show that the set $n$ of $x \in A$ such that $sx = 0$ for some $s \in S$ is an ideal of $A$. Set $A_n = A/n$ and denote by $S_1$ the canonical image of $S$ in $A_n$. Show that no element of $S_1$ is a divisor of 0 in $A$, and that the canonical homomorphism $S^{-1}A \to S_1^{-1}A_1$ is bijective.

Deduce that, for $S^{-1}A$ to be a finitely generated $A$-module, it is necessary and sufficient that all the elements of $S_1$ be invertible in $A$, in which case $S^{-1}A$ is identified with $A$.