Exercise 6, § 3

(a) Let M be the Z-module defined in Algebra, Chapter VII, § 3, Exercise 22 (h); let T be the torsion submodule of M. Show that, for every prime number $p$, the submodule $T_{(p)}$ of $M_{(p)}$ is a direct factor of $M_{(p)}$, although T is not a direct factor of M.

(b) Let M be the Z-module defined in Algebra, Chapter VII, § 3, Exercise 5; show that, for every prime number $p$, $M_{(p)}$ is a free $\mathbf{Z}_{(p)}$-module, although M is not a free Z-module.