I have to find a maximal ideal of $\mathbb{Z} \times \mathbb{Z}$ , and a prime ideal that is NOT maximal.
Or, essentially, I want $I$ such that $\mathbb{Z} \times \mathbb{Z} / I$ is a field,
and I want $P$ such that $\mathbb{Z} \times \mathbb{Z} / P $ is a domain but NOT a field.
What's my general strategy for something like this??
Best Answer
Hint 1: What are the maximal ideals in $\mathbb{Z}$? What about the prime ideals in $\mathbb{Z}$? Use this to help you find your answer.
Hint 2: Maximal ideals are always prime ideals (as you seem to already know), so if you have an idea of what the maximal ideals are, the prime ideals that are not maximal should be slightly smaller in some sense...