Exercise 5, § 2

Let L be a finitely generated free $\mathbf{Z}$-module and T a finite commutative group whose order is a power of a prime number p. Show that the canonical primary decomposition (Exercise 4) of {0} in $M = L \oplus T$ is $\{0\} = p^n L \cap T$, where n is the least integer $\geq 0$ such that $p^n T = 0$ (note that $p^n M = p^n L$).