Finite quotients of the integers localized away from one prime
= Finite quotients of the integers localized away from one prime
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 $A_{\neg p}/p^nA_{\neg p}\cong\mathbb Z/p^n\mathbb Z$.