I encountered the following question: (This is taken the book Module Theory - an approach to linear algebra, by T.S Blyth. This is question is from exercise 4.1)
Let $M=Rx$ be a cyclic $R$-module. Show that $M \cong R/Ann_R(x)$ (where $Ann_R(x)=\{\lambda\in R: \lambda x=0\}). $
Deduce that if $R$ is a PID (with $R$ being commutative and with 1) and if $Ann_R(x)=p^kR$ for some $p\in R$, then the only submodules of $M$ are those in the chain
$0=p^kM\subset p^{k-1}M\subset ...\subset pM \subset p^0M=M$.
The first part is clear to me. I also established the fact that there is a correspondence between the submodules of $M$ and the submodules of $R/p^kR$ which are of the form $tR/p^kR$ where $p^kR \subseteq tR \subseteq\ R$. The part I could not show is why the ideal $tR$ need to be of the form $p^vR$ for $0\leq v \leq k$. Maybe the general direction I thought I need is not good. Any help would be appreciated.