Prove that $\mathbb{Z \times Z}/ \{(3m,n)\in\mathbb{Z \times Z}:m,n \in \mathbb{Z}\} $ is a field.

Question

Let $R=\mathbb{Z \times Z}$ be a ring, and $I=\{(3m,n)\in\mathbb{Z \times Z}:m,n \in \mathbb{Z}\} $, $J=\{(3m,0)\in\mathbb{Z \times Z}:m \in \mathbb{Z}\} $ two ideals of $R$. Prove that

1. $R/I$ is a field,
2. $R/J$ is not a field.

My thought:

1. $$ R/I = \mathbb{Z \times Z} / \langle3\rangle\times\mathbb{Z} \cong( \mathbb{Z}/\langle3\rangle) \times (\mathbb{Z / Z}) \cong \mathbb{Z_3} \times \{0\} \cong \mathbb{Z_3}$$ and the last one is a field, so $R/I$ is also a field.

2. $$R/J= \mathbb{Z \times Z} / \langle3\rangle\times\{0 \} \cong( \mathbb{Z}/\langle3\rangle) \times (\mathbb{Z}/ \{ 0\}) \cong \mathbb{Z_3} \times \mathbb{Z} $$ and the lastone is not a field, so$R/J$ is not a field.

Are these answers correct?

Answer 1

1) Consider the ring homomorphism $$ \mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}/3\mathbb{Z} $$ defined by $(m,n)\mapsto m+3\mathbb{Z}$. What is its kernel? Is it surjective?

2) Consider the ring homomorphism $$ \mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z} $$ defined by $(m,n)\mapsto (m+3\mathbb{Z},n)$. What is its kernel? Is it surjective?