While trying to find a maximal ideal in $\mathbb{Z} \times \mathbb{Z}$, I ran into something that seems contradictory.
If we let $J = \{(a,a):a \in \mathbb{Z}\}$ then $J \cong \mathbb{Z}$ given the by isomorphism $(a,a) \mapsto a$.
Then (at least it seems to me):
$$J/(7,7)J \cong \mathbb{Z}/7\mathbb{Z}$$
This is a field, which implies $(7,7)J = \{(7a,7a):a \in \mathbb{Z}\}$ is a maximal ideal.
But $(7,7)J \subsetneq \{(7a, 7b): a,b \in \mathbb{Z}\}$ which is a counterexample to the claim that $(7,7)J$ is maximal.
Where have I gone wrong?
Best Answer
What you have written is correct: $J/(7,7)J$ is isomorphic to $\Bbb Z/7\Bbb Z$ but what I think you're doing is you're confusing $J/(7,7)J$ with $\Bbb Z^2/(7,7) J$. $(7,7)J$ is maximal in $J$. It is not maximal in $\Bbb Z^2$ as you have demonstrated.