We want to find all $n>0$ such that $\mathbb{Z}/n\mathbb{Z}$ has a $\mathbb{Z}[i]$-module structure.
This is what I have. First of all we have that for $k\in \mathbb{Z}$ we have that for $m\in \mathbb{Z}/n\mathbb{Z}$, $k\cdot m=[km]\in \mathbb{Z}/n\mathbb{Z}$.
Note that if we manage to answer what $i\cdot 1$ is equal to then we would be done since $(a+bi)m=am+b(im)$.
Say $i\cdot 1=x$. Then we have that $i\cdot x=i\cdot(1+...+1)=i\cdot 1+...+i\cdot 1=x+...+x=x^2$. That is, $i\cdot (i\cdot 1)=x^2$On the other hand, $(i\cdot i)\cdot 1=(-1)\cdot 1=-1$So we want $x^2=-1$. So any $n$ that satisfies $\mathbb{Z}/n\mathbb{Z}$ such that $-1$ is a square would work because we can define $i\cdot 1=x$ where $x$ is such that $x^2=-1$. Is the above correct?
Thanks.