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$T$ is a set of some prime numbers.S numbers. $S$ is the multiplictive set generated by T. $T$. How to compute the pontryagin Pontryagin dual of S^{-1}Z(Z $S^{-1}Z$ ($Z$ is integal ring and S^{-1}Z $S^{-1}Z$ is localization of Z $Z$ at S) $S$). If T $T$ is an empty set,then set, then the pontryagin Pontryagin dual of S^{-1}Z=Z $S^{-1}Z=Z$ is R/Z. $R/Z$. If T $T$ is the set of all prime numbers, the pontryagin Pontryagin dual of S^{-1}Z=Q $S^{-1}Z=Q$ is natruely naturally isomorphic to A_Q/Q,where A_Q $A_Q/Q$, where $A_Q$ is the adele ring of rational number Q. $Q$.
Question: I guess the pontryagin Pontryagin dual of S^{-1}Z $S^{-1}Z$ is isomorphic to (R $(R \times\RestrictedProduct_{p\in T}Q_{p})/(S^{-1}Z)times\prod_{p\in T}Q_{p})/(S^{-1}Z)$. Is it right?
