Definition 3

Let $M$ and $N$ be two $A$-modules and $f : M \to N$ a homomorphism. $f$ is called pseudo-injective (resp. pseudo-surjective pseudo-zero) if $\mathrm{Ker}(f)$ (resp. $\mathrm{Coker}(f)$, $\mathrm{Im}(f)$) is pseudo-zero; $f$ is called pseudo-bijective if it is both pseudo-injective and pseudo-surjective.

A pseudo-bijective homomorphism is also called a pseudo-isomorphism.

Suppose that $M$ and $N$ are finitely generated; then, for $f : M \to N$ to be pseudo-injective (resp. pseudo-surjective, pseudo-zero), it is necessary and sufficient that, for all $p \in P \cup \{\{0\}\}$, $f_p : M_p \to N_p$, be injective (resp. surjective, zero); this follows from the flatness of the $A$-module $A$, (cf. Chapter I, § 2, no. 3, Remark 2).

Example (3) Let $M$ be a torsion-free finitely generated $A$-module; then the canonical mapping $c_M : M \to M^{**}$ of $M$ to its bidual is a pseudo-isomorphism. For $M$ is identified with a lattice of $V = M \otimes_A K$ (no. 1, Proposition 1); we have seen that $M_p = M_p^{**}$ for all $p \in P$ (no. 2, Example 2) and, for $p = 0$, $M_p$ and $M_p^{**}$ are both equal to $V$.