Exercise 25, § 5

Let $A$ be a ring and $S$ a multiplicative subset of $A$. An ideal $a$ of $A$ is called $S$-invertible if there exists $s \in S$ and an ideal $b$ of $A$ such that $ab = As$; two ideals $a, b$ of $A$ are called $S$-equivalent if there exists $s, t$ in $S$ such that $sa = tb$. Show that the $S$-equivalence classes of the $S$-invertible ideals of $A$ form a multiplicative group $\mathfrak{C}_S$. If $S$ is saturated (§ 2, Exercise 1) and no element of $S$ is a divisor of 0 in $A$, show that the group $\mathfrak{C}_S$ is isomorphic to the classes of invertible sub-$S^{-1}A$-modules of the total ring of fractions $B$ of $A$ (and $S^{-1}A$).