Show that these rings of Gaussian integers are ideals in $\mathbb{Z}[i]$?

Question

Consider the ring of Gaussian integers: $\mathbb{Z}[i]$ = {a + bi | a, b ∈ Z} ⊂ $\mathbb{Q}[i]$ with $i^2$ = −1. Let I = $(2+3i)$ and J =$(2−3i)$. Show that I and J are ideals of $\mathbb{Z}[i]$.

Answer 1

This should be a comment, but reputation lacks. In general if $R$ is a ring and $r\in R$ then $(r)$ denotes the (principal) ideal generated by $r$. So in fact $I$ and $J$ mentioned in your question are ideals by definition. Maybe they want you to construct two ringhomomorphisms having $I$ and $J$ as kernels.