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My assignment was to find all of the prime ideals of $\mathbb{Z}_3 \times \mathbb{Z}_4$ so I first found all of the ideals which are:

$\mathbb{Z}_3 \times {0}$, ${0} \times \mathbb{Z}_4$, $\mathbb{Z}_3 \times \mathbb{Z}_2$,${0} \times \mathbb{Z}_2$

I then just modded the whole group by each of the ideals (ignoring the trivial cases). Which told me that

${0} \times \mathbb{Z}_4$, and $\mathbb{Z}_3 \times \mathbb{Z}_2$ where both maximal and prime.

Is there an easier way to do this by just looking at the structure of the ring? It feels like the way I approached this question was much slower than it should have been.

Mathematical Dummy
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1 Answers1

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First of all, when you're talking about ideals, you've got rings, not groups.

Second, in $\Bbb Z_3×\Bbb Z_4\cong \Bbb Z_{12}$, the prime ideals are the prime ideals in $\Bbb Z$ containing $(12)$, by the correspondence theorem.

So we get just $(12)\subset(2),(3)\subset \Bbb Z$, which correspond to $(\bar2),(\bar3)\subset \Bbb Z_{12}$. These are $\{(2,2),(1,0),(0,2),(2,0),(1,2),(0,0)\}, 0×\Bbb Z_4$ (after isomorphism).

So, note: $\Bbb Z_3×\Bbb Z_2$ is wrong. It contains $(1,1)$, so any ideal containing it is the whole ring.

You can also tell because the respective quotients, $\Bbb Z_2,\Bbb Z_3$ are integral domains.

calc ll
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