Give an example of an ideal in $\mathbb{Z}\times\mathbb{Z}$ which is maximal.

Question

My answer right now is just $(0,1)$ and $(1,0)$ resulting in $\mathbb{Z}\times\mathbb{Z}$ as $(1,1)$. But this is the entire ring... Help?

Answer 1

If you want to check if your ideal is maximal, let $R$ be a ring and let $M$ be your maximal ideal. If ${R}/{M}$ is a field, then $M$ is maximal.

I gave you a hint to think of primes as a general case. For example $\mathbb{Z}$ x $\mathbb{2Z}$ is an example of a maximal ideal in your case.