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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.

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Facts you can prove that will help you decide. (They are not all directly related to the final answer, but they are related to other things you asked.)

  1. The product of two principal ideal rings is a principal ideal ring.

  2. This ring is not a domain.

  3. This ring is finite, and every prime ideal in a finite ring is maximal.

  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 $S$.

Here is another question that you would probably like to read.

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