Exercise 25, § 5

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