Proposition 7

Let $A, B$ be two rings and $\rho$ a homomorphism of $A$ to $B$. Let $E$ be a faithfully flat right $B$-module. For $\rho^*(E)$ to be a flat (resp. faithfully flat) right $A$-module, it is necessary and sufficient that $B$ be a flat (resp. faithfully flat) right $A$-module.

Applying Proposition 4 of no. 2 with $B, A, E, B$ in place of $R, S, E, F$ respectively, and the right $A$-module structure on $B$ being defined by $\rho$, it is seen that

B is a flat (resp. faithfully flat) A-module if and only if $E \otimes_B B = \rho^*(E)$ is a flat (resp. faithfully flat) A-module.

Remarks (1) Proposition 7 shows that, for B to be a faithfully flat A-module, it is sufficient that there exist one faithfully flat B-module which is also a faithfully flat A-module. (2) Let A, B, C be three rings and $\rho : A \to B, \sigma : B \to C$ two ring homomorphisms. Proposition 7 shows that if C is a faithfully flat B-module and B a faithfully flat A-module, then C is a faithfully flat A-module. If C is a faithfully flat B-module and a faithfully flat A-module, then B is a faithfully flat A-module (taking the modules as right modules, to fix the ideas). On the other hand B and C may be faithfully flat A-modules without C being a faithfully flat B-module (Exercise 7).