Chapter 6.4, Problem 2E

Let $R$ be as in Exercise $1$, and show that the principal ideal $I=\left(\sqrt{2}\right)=\left\{2n+m\sqrt{2}|n\in \mathbb{Z},m\in \mathbb{Z}\right\}$ is a maximal ideal of $R$.

Exercise 1.

According to part $a$ of Example $3$ in Section $5.1$, the set

$R=\left\{m+n\sqrt{2}|m\in \mathbb{Z},n\in \mathbb{Z}\right\}$

is a ring. Assume that the set

$I=\left\{a+b\sqrt{2}|a\in E,b\in E\right\}$

is an ideal of $R$, and show that $I$ is not a maximal ideal of $R$.