TITLE: Maximal ideals in $\mathbb Z[i]$
QUESTION [6 upvotes]: Let $\mathbb Z[i]=\{a+bi \mid a,b∈ℤ\}$ be a subring of $ℂ$. 
Consider two principal ideals $I=(7)$ and $J=(13)$ in $\mathbb Z[i]$. Is the ideal $I$ maximal? Is the ideal $J$ maximal?

REPLY [6 votes]: Hint:
A principal ideal $(a)$ in a PID $R$ is maximal iff $a$ is irreducible.