Chapter 6.1, Problem 29E

Let $G$ be the set of Gaussian integers $\left\{m+ni|m,n\in \mathbb{Z}\right\}$. Let $I=\left\{a+bi|a\in \mathbb{Z},b\in {\rm E}\right\}$.

a. Prove or disprove that $I$ is a substring of $G$.

b. Prove or disprove that $I$ is an ideal of $G$.