Proposition 9

Let $A, B$ be two rings and $\rho$ a homomorphism of $A$ to $B$. The following five properties are equivalent: (a) The right $A$-module $B$ is faithfully flat. (b) The homomorphism $\rho$ is injective and the right $A$-module $B/\rho(A)$ is flat. (c) The right $A$-module $B$ is flat and, for every left $A$-module $F$, the canonical homomorphism $x \mapsto 1 @ x$ of $F$ to $B @ * F$ is injective. (d) The right $A$-module $B$ is flat and, for every left ideal $a$ of $A$, $\rho^{-1}(\mathrm{Ba}) = a$. (e) The right $A$-module $B$ is flat and, for every maximal left ideal $m$ of $A$, there exists a maximal left ideal $n$ of $B$ such that $\rho^{-1}(n) = m$.

By Proposition 8, (a) implies each of the properties (c), (d), (e). On the other hand, if (e) holds, then $Bm \# B$ for every maximal left ideal $m$ of $A$ (since there exists a maximal left ideal $n$ of $B$ such that $Bm \subset n$), and $B$ is a faithfully flat $A$-module by criterion (d) of Proposition 1 of no. 1; hence (e) implies (a).

We shall now prove that (c) $\Rightarrow$ (d) $\Rightarrow$ (b) $\Rightarrow$ (a), which will complete the proof. In the first place, (c) implies (d) by taking $F = A_s/a$ in (c). If (d) holds, by taking $a = \{0\}$ it follows that $\rho$ is injective; (d) and § 2, no. 6, Corollary to Proposition 7 imply that $B/\rho(A)$ is a flat right $A$-module, that is (d) implies (b). Finally, if (b) holds, Proposition 3 of no. 1 applied to the exact sequence

shows that $B$ is a faithfully flat right $A$-module, since $A_a$ is faithfully flat.

*Remark (2). If $A$ and $B$ are commutative, we shall see in Chapter II, § 2 no. 5, Corollary 4 to Proposition 11 that the conditions of Proposition 9 are equivalent to the following:

(f) B is a flat A-module and, for every prime ideal $\mathfrak{p}$ of $A$, there exists an ideal $q$ of $B$ such that $\bar{\rho}^{-1}(q) = \mathfrak{p}$.

Under the conditions of Proposition 9, let us identify $A$ with a subring of $B$ by means of $\mathfrak{p}$. The relation $\bar{\rho}^{-1}(Ba) = a$ then reads $A \cap Ba = a$. On the other hand if $F$ is a left $A$-module, $F$ is identified with its image in $B \otimes_A F$ under the canonical mapping $x \mapsto 1 @ x$; if $X$ is an additive subgroup of $F$, then we denote by $BX$ the left sub-B-module of $B \otimes_A F$ generated by $X$. With this notation, we have: