I have elements of the form $a+bi$. I have attempted to consider arbitrary ideals in $\mathbb{Z}[i]$. If $N$ is ideal and $N=\{0\}$ then it is generated by $0$.
If $N$ is not trivial, then exists $f=a+bi$ in $N$. But from here I do not see which direction I need to proceed to show that ideal needs to be principal... Also I don't see any units except $1$. I believe I am missing something very important. Thanks in advance for any help!