$M$ is a $D-module$ torsion and finitely generated. Then $M$ is cyclic if and only if $\exists p_1, \dots, p_r\in D$ not associated and $n_1,\dots, n_r\in\mathbb{Z}$ such that $M\simeq\bigoplus_{i=1}^r D/\langle p_i^{n_i}\rangle$.
(0) I have to point out that I find confusing the statement that the module is fintely generated and then add the condition "$M$ is cyclic". If $M$ is cyclic then it is obviosly finitely generated.
(1) Let's assume $M$ is cyclic, then it is generated by a single element. Set $M=\langle a \rangle$. This means $\bigoplus_{i=1}^r D/\langle p_i^{n_i}\rangle$ should be generated by a single element if these two sets are to be isomorphic (thinking about defining the isomorphism between the generating sets, but I don't think it is possible so I'm not going there).
How to use the torsion condition? I know it means $mr=0, m\in M$ for some $ r\in D$.
(2) I am particularly confused on what the set $\langle p_i^{n_i}\rangle$ is as it looks like a combinations of primes generating a number.
(3) Considering the isomorphism theorem: If $M$ is finitely generated then $M\simeq \bigoplus D/s_i$. (What are the $s_i$ elements here?). This text http://math.uchicago.edu/~womp/2007/PIDmod2007.pdf gives a construction that is similar to the isomorphism I want to show, but I don't see how being cyclic and torsion plays a part in the argument.
