Exercise 9, § 2

a) Let $M$ be a $n$-primary module, and let $M_1$ be the submodule of $M$ consisting of elements of infinite height in $M$. Show that the $n$-primary module $M/M_1$ contains no nonzero element of infinite height.

b) Let $p$ be a prime number, and let $E$ be the direct sum of the $p$-groups $E_n = \mathbf{Z}/(p^n)$ ($n \geq 1$); let $e_n$ denote the residue class of 1 mod $p^n$ in $E_n$. Let $H$ be the submodule of $E$ generated by the elements $e_1 - p^{n-1} e_n$ for all integers $n \geq 2$. Let $M$ be the quotient module $E/H$. Show that the submodule $M_1$ of elements of infinite height in $M$ is the submodule $(E, + H)/H$, which is isomorphic to $E_1$, and show that there is no nonzero element in $M_1$, which has infinite height in $M$.