Exercise 2, § 5

Let M be an A-module. One says that a submodule M’ of M is pure if the complex defined by the exact sequence $0 \to M' \to M \to M/M' \to 0$ is pure (exercise 1).

(a) Now prove that when A is a principal ideal domain, the notion of pure submodule coincides with that of VII, § 2, exercise 7.

(b) Suppose that the A-module M is flat. Now prove that the following conditions are equivalent:

α) the submodule M’ is pure ;

β) the quotient $M/M'$ is flat ;

γ) for every ideal a of A, one has $M' \cap aM = aM'$.