# What are the prime and maximal ideals of $\mathbb{Z}_4 \times \mathbb{Z}_4$?

I know I have to find all the subgroups of $\mathbb{Z}_4 \times \mathbb{Z}_4$ then check which ones are ideals, and I got 2 principal ideals $\langle (3,2) \rangle$ and $\langle (2,3) \rangle$. But then I forgot that $\mathbb{Z}_4 \times \mathbb{Z}_4$ isn't a principal ideal domain (or at least I haven't proved it), and there should be one more prime and max ideal that isn't principal.

But I'm having trouble finding subgroups of $\mathbb{Z}_4 \times \mathbb{Z}_4$.

Also, what is $\mathbb{Z}_4 \times \mathbb{Z}_4 / I$ isomorphic to for every such ideal? I couldn't think of any other way than brute-force thinking of all the elements and finding a suitable isomorphism.

4. The ideals of $R\times S$ are of the form $A\times B$ where $A$ is an ideal of $R$ and $B$ is an ideal of $S$, and the maximal ideals are of the form $M\times S$ and $R\times N$ where $M$ is a maximal ideal of $R$ and $N$ is a maximal ideal of $S$.