A direct sum of cyclic modules over a PID

Question

Let $R$ be a PID, let $p$ be a prime of $R$, and let $M$ be the $R$-module $R/Rp^{e_1}\oplus \cdots \oplus R/Rp^{e_n}$ where the $e_i$ and $n$ are positive integers. Define $M(p)=\{m: pm=0\}$ and $pM=\{pm:m\in M\}$. Then show that $M/pM \cong M(p)$.

I couldn't able to define a surjective $R$-module homomorphism from $M$ to $M(p)$. Any help would be great.

Answer 1

Consider first the case $M=R/Rp^k$ and the map $m\mapsto p^{k-1}m$.