Remark (2)

In the statement of Theorem 5, the modules $A/p_i^{n_i}$ may be replaced by $A/p_i^{(n_i)}$ ($§ 1$, no. 4, Proposition 8). For all $p \in P$, the canonical mapping $g : A/p^n \to A/p^{(n)} = A/(A \cap p^nA_p)$ is a pseudo-isomorphism, as, for all $q \in P$ distinct from $p$, $A_q/p^nA_q = A_q/p^{(n)}A_q = 0$ and $A_p/p^nA_p = A_p/p^{(n)}A_p$.

\* Given an exact sequence of $A$-modules, $E \to F \to G$, if $E$ and $G$ are pseudo-zero, so is $F$, as follows from Definition 2 and Chapter II, §2, no. 4, Theorem 1. In the language of categories, we may then say that, in the category $\mathscr{C}$ of $A$-modules, the sub-category $\mathscr{C}'$ of pseudo-zero modules is full and we may then define the quotient category $\mathscr{C}/\mathscr{C}'$: the objects in this category are also $A$-modules but the set of morphisms from $E$ to $F$ (for $E, F$ in $\mathscr{C}/\mathscr{C}'$) is the direct limit of the set of commutative groups $\mathrm{Hom}_A(E', F')$, where $E'$ (resp. $F'$) runs through the set of submodules of $E$ (resp. the set of quotient modules $F/F''$ of $F$) such that $E/E'$ (resp. $F''$) is pseudo-zero. Of course, for every ordered pair of $A$-modules $E, F$, there is a canonical homomorphism $\mathrm{Hom}_{\mathscr{E}}(E, F) \to \mathrm{Hom}_{\mathscr{E}/\mathscr{E}'}(E, F)$. To say that a homomorphism $u \in \mathrm{Hom}_A(E, F)$ is pseudo-zero (resp. pseudo-injective, pseudo-surjective, pseudo-bijective) means that its canonical image in $\mathrm{Hom}_{\mathscr{E}/\mathscr{E}'}(E, F)$ is zero (resp. a monomorphism, an epimorphism, an isomorphism). \*