Exercise 15, § 3

Let A be a (commutative) ring, M an A-module.

a) In order that a submodule $M'$ of M be pure (Chapter I, § 2, Exercise 24), it is necessary and sufficient that, for every maximal ideal $m$ of A, $M'_m$ be a pure sub-$A_m$-module of $M_m$ (use Theorem 1 of No. 3).

b) Suppose that A is a local ring with maximal ideal $m$, and M a free a-module of finite type. In order that a submodule $M'$ of finite type of M be pure, it is necessary and sufficient that it be a direct factor of M. (Using Corollary 1 of Proposition 5 of No. 2, reduce to proving that, if $M' \subset mM, M'$ can only be a pure submodule of M if $M' = 0$.)