Proposition 8

Let $A, B$ be two rings, $E$ a right $A$-module and $F$ an $(A,B)$-bimodule. Suppose that $E$ is flat and that $F$ is flat as a $B$-module. Then the $B$-module $E \otimes_A F$ is flat.

Let $G$ be a left $B$-module and $G'$ a submodule of $G$. Since $F$ is flat as a right $B$-module, the canonical homomorphism $F \otimes_B G' \to F \otimes_B G$ is injective. Since $E$ is flat, the canonical homomorphism

is injective. As $E \otimes_A (F \otimes_B G')$ and $E \otimes_A (F \otimes_B G)$ are canonically identified with $(E \otimes_A F) \otimes_B G'$ and $(E \otimes_A F) \otimes_B G$ respectively (Algebra, Chapter II, § 3, no. 8, Proposition 8), the canonical homomorphism

is injective, which proves that $E \otimes_A F$ is a flat $B$-module.