Let $R=\mathbb{Z}[i]$ be the ring of Gaussian integers. Let $z=5 + 3i$ and let $I=\left< z \right>$. Let $\phi: \mathbb{Z} \rightarrow R/I$ be the canonical ring homomorphism.
I am trying to show that $\phi$ is surjective. I understand that working in $R/I$ we have $5+3i=0$. I also understand that $\phi(c) = 1 + 1 + ... + 1$ ($c$ times). My problem is that I cannot see how to find a $c$ such that $\phi(c) = 1+i$, for example, because $1+i$ is not a multiple of $5+3i$. And how can I think of $\phi(c)$ when $c$ is negative?
How do I proceed? Any pointers are much appreciated.
Note that it is enough to check that there exists some $c$ such that $\phi(c)=i+I$ since $1$ and $i$ generate $\mathbb{Z}[i]$. Since $5$ and $3$ are coprime, there exist integers $a,b$ such that $5b+3a=-1$. Letting $c=5a-3b$ yields $c-i=(5+3i)(a+bi)$, i.e. $\phi(c)=i+I$.