What is the (natural) bijection between the isomorphic class of sub modules and isomorphic class of quotient modules of a finitely generated torsion module over a PID.
Is there any inclusion relation between these classes?



Correspondence between submodules and quotient modules
If $M=R/(p^n)$ with $p$ prime, the result is clear. Since an arbitrary $M$ is a direct sum of such modules, the result is still clear.


