Factor Ring question and finding maximal ideals of $\mathbb{Z}\times\mathbb{Z}$

Question

What is the maximal ideal of $\mathbb{Z}\times\mathbb{Z}$?

I think since $(\mathbb{Z}\times\mathbb{Z})/(\{0\}\times\mathbb{Z})$ is isomorphic to $\mathbb{Z}$, it seems like that

$(\mathbb{Z}\times\mathbb{Z})/(\{0\}\times p\mathbb{Z})$ is isomorphic to $\mathbb{Z}_p$ for $p$ prime.

Then $\mathbb{Z}_p$ is a field and thus $\{0\}\times\mathbb{Z}_p$ is a maximal ideal.

In my book, it says that $(\mathbb{Z}\times\mathbb{Z})/(\mathbb{Z}\times p\mathbb{Z})$ is isomorphic to $\mathbb{Z}_p$ and therefore $\mathbb{Z}\times p\mathbb{Z}$ is a maximal ideal.

Isn't $(\mathbb{Z}\times\mathbb{Z})/(\{0\}\times p\mathbb{Z})$ isomorphic to $\mathbb{Z}_p$?

Answer 1

Note that in general you have $$ \frac{A\times B}{I\times J}\simeq\frac AI\times\frac BJ. $$ Since a product of two non-trivial rings is not a domain the ideal $I\times J$ is not even a prime ideal unless $I=A$ or $J=B$.

Answer 2

A maximal ideal of $\mathbb{Z}\times\mathbb{Z}$ must be of the form $I\times\mathbb{Z}$ or $\mathbb{Z}\times J$ where $I$ and $J$ are maximal ideals of $\mathbb{Z}$. For example $p\mathbb{Z}\times \mathbb{Z}$ is a maximal ideal if $p$ is a prime.

$(\mathbb{Z}\times\mathbb{Z})/(\{0\}\times p\mathbb{Z})$ is isomorphic to $\mathbb{Z}\times \mathbb{Z}_p $ not to $\mathbb{Z}_p$.