Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$.

For any $q_1, q_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ such that $q_2 = n \cdot q_1$, we can form a surjective homomorphism $\Bbb Q/q_2\Bbb Z \to \Bbb Q/q_1\Bbb Z$.

The set of all such $\Bbb Q/q\Bbb Z$ forms a poset with these surjections, and we can take the inverse limit to get something like a "rational solenoid."

Is this group isomorphic to the finite adele ring $\Bbb A_\Bbb Q^f \cong \Bbb Q \otimes \hat{\Bbb Z}$?

Is this group equal to the same thing you'd get if you took the inverse limit of $\Bbb Q/n\Bbb Z$ for a natural number $n$ instead of a positive rational $q$?

For #1, I think it is, because the dual group should be a direct limit of localizations of $\hat{\Bbb Z}$, which I believe is isomorphic to $\Bbb Q \otimes \hat{\Bbb Z}$, which is self-dual. The same reasoning applies for #2.