Take the 2-minute tour ×  I posed the question here, but get no
answers yet.  Let $R$ be a PID, $M$ be an $R$-module. If $M$ is
isomorphic to $r$ copies of cyclic primary module $R/\langle
p^s\rangle$ where $p$ is a prime element of $R$, then does $M$
possess the following property?  Given any submodule $N$ of $M$
isomorphic to $R/\langle p^{s_1}\rangle\oplus\cdots\oplus
R/\langle p^{s_r}\rangle$, $M/N$ is isomorphic to $R/\langle
p^{s-s_1}\rangle\oplus\cdots\oplus R/\langle p^{s-s_r}\rangle$.
share|improve this question This is better suited for MSE. You
could start with $r=1$, with $M=R/p^sR$, $N=R/p^{s_1}R$. – 
Dietrich Burde Jun 26 '13 at 8:23  1 Answer 1  up vote 3 down
vote accepted  Choose generators $n_1,\dotsc,n_r$ for $N$ with
$p^{s_i}n_i=0$. It is not hard to see that the annihilator of
$p^{s_i}$ on $M$ is $p^{s-s_i}M$, so we can choose $m_i$ with
$p^{s-s_i}m_i=n_i$. If we can prove that the elements $m_i$ form
a basis for $M$ over the ring $R/p^s$, then everything else is
clear.  The given assump