Finitely generated modules, related to the Gaussian integers

Question

Let $M = \mathbb{Z}[i] = \{a + bi|a, b \in \mathbb{Z}\}$ be the additive group of the Gaussian integers.

Consider the $\mathbb{Z}$-submodules $N_1 = 2M$, $N_2 = (1 + i)M$.

I now want to figure out the structure of the $\mathbb{Z}$-modules $M/N_1$, $M/N_2$.

Thanks in advance. I thought this could maybe be done via the structure theorem for finitely generated modules over a principal ideal domain? I'm not exactly sure how though.

Answer 1

Note that as $\mathbb{Z}$-modules, $M\cong\mathbb{Z}\oplus\mathbb{Z}$, $N_1\cong\{(a, b)\in\mathbb{Z}\oplus\mathbb{Z}| a\text{ and }b\text{ are both even}\}$ and $N_2\cong\{(a, b)\in\mathbb{Z}\oplus\mathbb{Z}|a+b\text{ is even}\}$. So $M/N_1\cong\mathbb{Z}_2\oplus\mathbb{Z}_2$ and $M/N_2\cong\mathbb{Z}_2$.

Answer 2

Hint 1: $2=(1+i)(1-i)$ and $1\pm i$ is irreducible. Use the Chinese Remainder Theorem.

Hint 2: $M/N_1$ has order $4$ and exponent $2$. What abelian group that has that?