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I want to show that every ideal in $\mathbb{Z} \times \mathbb{Z}[i]$ is principal, find the prime ideals and the maximal ideals in this ring. Can anyone give me a hint or suggestion on how to attempt this problems? I always feel lost when I see them and don't even know where to start. Thanks for your help!

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  • $\begingroup$ It is also good to know that $\mathbb{Z} \times \mathbb{Z}[i]$ is not a principal ideal domain. $\endgroup$ – Bumblebee Dec 19 '17 at 18:19
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Hint: Given two commutative rings with unity $A$ and $B$, the ideals of $A \times B$ are all of the form $I \times J$.

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There is a general theorem saying: if $A,B$ are rings (say commutative rings with unit, to make things easier), then any ideal of $A\times B$ has the form $I\times J,$ where $I$ is an ideal of $A$ and $J$ is an ideal of $B$.

Assuming for a second this result is proved, then you should be able to show easily that if every ideal of $A$ is principal, and every ideal of$B$ are principal, then the same holds for $A\times B$.

To prove the general theorem (you can do it in your particular case if you want to), I would suggest to take an ideal $M$ of $A\times B$, and to consider $I=\{ a\in A \mid (a,0)\in M\}$ and $J=I=\{ b\in B \mid (0,b)\in M\}$, and show that $I$ is an ideal of $A$, $J$ is an ideal of $B$, and $I\times J=M$.

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