Correspondence between submodules and quotient modules What is the (natural) bijection between the set of all sub modules upto isomorphism and set of all isomorphic quotient modules upto isomorphism of a finitely generated torsion module over a PID. Is there any inclusion relation between these classes?
 A: If $M$ is a f.g. torsion module over a PID $R$, one can show that every submodule of $M$ is isomorphic to a quotient of module of $M$, and vice versa, by looking at a decomposition of $M$ into direct summands of cyclic modules. However, this correspondence is not natural.
However, the following works: Let $K$ be the field of fractions of $R$. Then $\hom(-,K/R)$ ("Pontrjagin dual") is a contravariant functor from $R$-modules to $R$-modules. For every $R$-module $M$ there is a canonical homomorphism $M \to \hom(\hom(M,K/R),K/R)$. If $M$ is finite cyclic, this is easily seen to be an isomorphism. Hence, it is an isomorphism for all f.g. torsion modules. It follows that $\hom(-,K/R)$ is an anti-equivalence of categories from f.g. torsion modules to itsself. 
Now it follows by abstract nonsense that submodules (quotients) of $M$ correspond naturally to quotients (submodules) of $\hom(M,K/R)$.
Besides, for every f.g. torsion module $M$ there is some isomorphism $M \cong \hom(M,K/R)$ (which is not natural). But once this is fixed, we get a correspondence between submodules of $M$ and quotients of $M$, where these modules are abstractly isomorphic.
