Exercise 13, § 1

Now prove that every finitely generated flat module P over an integral ring A is projective (consider a presentation $0 \to R \xrightarrow{i} L \xrightarrow{\rho} P \to 0$, where L is a free module of finite type; choose a finitely generated submodule R' of R such that $R/R'$ is torsion, and apply theorem 1, p. 14).