I think there are 3 ideal and maximal primes:
$\langle(0,1)\rangle$ since factor group over $\langle(0,1)\rangle$ is isomorphic to $\mathbb{Z}_2$, which is field and integral domain.
And same reason for $\langle(1,0)\rangle$ .
How about $\langle(1,1)\rangle$ ? I think factor group $(\mathbb{Z}_2 \times \mathbb{Z}_2) / \langle(1,1)\rangle$ is also isomorphic to $\mathbb{Z}_2$ and therefore $\langle(1,1)\rangle$ is also maximal and prime ideals. but answer key only states that $\langle(1,0)\rangle$ and $\langle(0,1)\rangle$ are answer.
Can anyone explain?
Best Answer
Note that $(1,1)$ is the identity element of the ring, so any ideal containing it contains the whole ring. You're quotienting out by the subgroup generated by $(1,1)$, which is not the same as the ideal generated by it.