Let $R= \{ a+bi : a,b \in \mathbb{Z} \}$ be a subring of $\mathbb{C}$. Consider two principal ideals $I=(7)$ and $J=(13)$ in $R$. Is the ideal $I$ maximal? How about $J$?
I don't understand what $I=(7)$ or $J=(11)$ means. Aren't there both $a$ and $b$ to be considered? Why is there only one number?
I know that an ideal $M$ in a ring $R$ is maximal if $M \ne R$ and whenever if $N$ is also an ideal such that $M \subseteq N \subseteq R$, then $M=N$ or $N=R$. So if I can understand what these ideals are I can probably figure out which one is maximal.
Any help will be greatly appreciated.
Thanks in advance!!
Best Answer
Hint
$\mathbb Z[i]$ is a PID. If $(7)$ or $(13)$ is not maximal, it can be included in some larger ideal $(m)$.
This means that $m|7$ or $m|11$ in $\mathbb Z[i]$. Now all you need is to figure out if you can find $a+bi$ and $c+di$ which are not units and
$$(a+bi)(c+di)=7$$ respectively $$(a+bi)(c+di)=13$$