Remark

In the former of the two above cases, the mapping $i : D(A) \to D(B)$ is injective: as the elements of $P(B)$ form a basis of $D(B)$ and two distinct ideals of $P(A)$ cannot be the traces on A of the same ideal of $P(B)$, it amounts to verifying that $i(p) \neq 0$ for all $p \in P(A)$; now, this follows from Chapter V, § 2, no. 1, Theorem 1. It is similarly seen that, if B is a faithfully flat A-module, $i$ is injective (Chapter II, § 2, no. 5, Corollary 4 to Proposition 11).

In what follows, we propose to study the canonical homomorphism $\tilde{i}$ from $C(A)$ to $C(B)$ for certain ordered pairs of Krull domains A, B.