Corollary 3

Let $R, S$ be two rings and $\phi : R \to S$ a ring homomorphism. If $M$ is a flat right $S$-module and $\phi_*(S_d)$ is a flat right $R$-module, then $\phi_*(M)$ is a flat right $R$-module.

Recall that $\phi_*(M)$ is the right $R$-module defined by $x.r = x.\phi(r)$ for all $x \in M$ and all $r \in R$ (Algebra, Chapter II, § 1, no. 13). Then apply Proposition 8 with $A = S, B = R, E = M$ and $F = S, S$ having the structure of a $(S, R)$-bimodule defined by $\phi$; the right $R$-module $M \otimes_S S$ is then precisely $\phi_*(M)$.