I need some help with this question:
Let $A$ be the subring of $\mathbb{Q}(i)$ generated by $\mathbb{Z}[i]$, $\frac{1}{1+2i}$ and $\frac{1}{2+3i}$. Given $n\in\mathbb{Z} \setminus \{0\}$, can we ensure that $A/nA$ is finite?
The elements of $A$ are elements in $\mathbb{Q}(i)$ with powers of $1+2i$ and $2+3i$ in their denominators, e.g. $$\frac{1}{(1+2i)^r(2+3i)^s}.$$
Any help will be really appreciated.
As I have one vote to close the topic, I'll add some extra information. This question is part of an investigation based on the Pyjama problem (http://arxiv.org/abs/1305.1514). I was working with a teacher and we think that $A/nA$ is not always finite, but I did the question here to know if anyone has a different opinion. :)
$A=S^{-1}R$, where $R=\mathbb Z[i]$ and $S$ is the multiplicative set generated by $1+2i$ and $2+3i$. Then $A/nA$ is isomorphic to a ring of fractions of $R/nR$ which is a finite ring.