Exercise 12, § 1

Let $\rho : A \to B$ be a homomorphism of rings which makes B a right A-module that is faithfully flat; let M be an A-module. a) Now prove that, in order that M be of finite type (resp. of finite presentation), it is necessary and sufficient that the same hold for the B-module $B \otimes_A M$. b) Suppose that $\rho(A)$ is contained in the center of B. Now prove that M is flat (resp. faithfully flat, resp. finitely generated projective) if and only if the B-module $B \otimes_A M$ is.