Flat Tensor Products (10.5.23)

Dummit and Foote Abstract Algebra, section 10.5, exercise 23:

When $M$ is a right flat $R$-module and $S$ is a ring considered as a left $R$-module by some identity-fixing homomorphism $R → S$, prove that $M \otimes_R S$ is a right flat $S$-module.

Proof: Let $0 → A → B$ be an exact sequence of $S$ modules by $ψ$. Since $S$ is a free right $S$-module of rank 1, it is flat, and therefore $1 \otimes_S ψ : S \otimes_S A → S \otimes_S B$ is injective. Moreover, this produces an exact sequence of $R$-modules, and since $M$ is right flat, $1 \otimes_R (1 \otimes_S) : M \otimes_R S \otimes_S A → M \otimes_R S \otimes_S B$ is injective, which is the associated homomorphism induced by the functor $M \otimes_R S \otimes_S \_$, which is to say $M \otimes_R S$ is a right flat $S$-module.$~\square$