Consider the multiplicative group of strictly positive rationals, denoted $\Bbb Q^+$. This can be viewed as a $\Bbb Z$-module, with the primes serving as a basis.
If we place the order topology on $\Bbb Q^+$, this turns it into a topological $\Bbb Z$-module.
Main question: Does this naturally induce a topology on the dual module $\text{Hom}(\Bbb Q^+, \Bbb Z)$?
Secondary question: Does this naturally induce a topology on $\Bbb Z$, making it a topological ring?