Remark

The hypothesis of Corollary 2 is fulfilled in the two following cases:

(1) $A$ is an integral domain and $S$ is the set of elements $\neq 0$ of $A$, $B$ then being the field of fractions of $A$. The invertible sub-A-modules of $B$ are also called in this case invertible fractional ideals; those which are monogenous free $A$-modules $Ab$ ($b \# 0$ in $B$) are just the fractional principal ideals defined in Algebra, Chapter VI, § 1, no. 5.

(2) The ring $A$ is Noetherian and $S$ is the set of elements of $A$ which are not divisors of 0 such that $B$ is the total ring of fractions of $A$. In this case $S = A - \bigcup p_i$, where the $p_i$ are the elements (finite in number) of $\mathrm{Ass}(A)$ (Chapter IV, § 1), hence $B$ is semi-local (§ 3, no. 5, Proposition 17).*