Chapter 6.4, Problem 12E

Let $R$ bethe ring of Gaussian integersas an Exercise 11, and let

$I=\{a+bi|2$ divides $a$ and $2$ divides $b\}$.

Show that $I$ is an ideal of $R$.

Show that $I$ is not a maximal ideal of $R$.

Exercise 11.

Let $R$ bethe ring of Gaussian integers $\left\{m+ni|m,n\in \mathbb{Z}\right\}$. Let

$M=\{a+bi|3$ divides $a$ and $3$ divides $b\}$.

Show that ${\rm M}$ is an idea of $R$.

Show that ${\rm M}$ is a maximal ideal of $R$.