Integer localization at a set of primes ($A_{\pi }$)

For a set $\pi$ of prime numbers, the integer localizationconsists of fractions whose denominator has prime factors only in $\pi$. As an additive group it is divisible by every prime in $\pi$.

If $A_{\neg p}=\mathbb{Z}[q^{-1}:q\ne p]$, its finite quotients are exactly the cyclic $p$-groups $\mathbb{Z}/p^n\mathbb{Z}$. Divisibility by every $q\ne p$ excludes other prime torsion, while a quotient of exponent $p^n$ factors through $A_{\neg p}/p^n{A}_{\neg p}\cong \mathbb{Z}/p^n\mathbb{Z}$.