Quotienting $\Bbb{Q}$ by a fractional ideal, what happens?

Question

Consider arithmetic $\pmod N, \ N \in \Bbb{N}$.

Now suppose there is $a \in \Bbb{Z}/N$ such that $(a,N) = 1$. Then Can we consider arithmetic in $R:=\Bbb{Q}/(N/a)\Bbb{Z}$ equivalent in some way to arithmetic in $\Bbb{Z}/N$?

If we're in $R$ then equality $x = y \iff (N/a) \mid (x - y) \iff (x-y)/(N/a) \in \Bbb{Z} \iff ax - ay \in N \Bbb{Z}$.

Thus you see there is a way to get back to $\Bbb{Z}/N$ and things are equal iff they are in $R$. So is $R$ finite?

Answer 1

For any non-zero rational $$\Bbb{Q/xZ\cong x^{-1}(Q/xZ)=x^{-1} Q/Z=Q/Z}$$ And $\Bbb{Z/nZ\cong n^{-1}Z/Z}$ is a finite subgroup of it.