Forgive me for this naive question. 

We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, [page 784](https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA784).

> 
> Every module is a submodule of an injective module.
> 
> 
> 

For a module $M$, Lang defines its dual to be $M$^ = $Hom(M, \mathbb{Q}/\mathbb{Z})$.

This "dual", if anything, is an algebraic version of the Pontrjagin dual of a topological group $G$, which is $Hom(G, \bf{S}^1)$, where we consider the continuous homomorphisms. $\mathbb{Q}/\mathbb{Z}$ is the torsion part of $\bf{S}^1$, and it seems this restriction is done for preventing too big a dual.

The proof goes on and shows that any module is a submodule of an injective module.

However, is there any conceptual explanation of why this works, and why this mysterious appearance of a dual which looks so much like the Pontrjagin dual?