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Let $R_1=\mathbb{Z}$ and $R_2=\mathbb{F}_4$.

1) Find all ideals in $R_1 \times R_2$.

Since $\mathbb{Z}$ has only principal ideals as $(0), (1), (2)...$

The ideals of $\mathbb{F}_4$ is where I get confused, I think they are also principal ideals which are $(0), (1), (2), (3)$.

Let $I_1 \rhd R_1$ and $I_2 \rhd R_2$. Then, the ideals of $R_1 \times R_2$ are

$I_1 \times I_2=\{((i_1),(i_2)): i_1 \in \mathbb{Z}, i_2 \in \{0,1,2,3\}\}$

2) Which of them are principal ideal?

I tend to think that they are all principal ideals.

3) Which of them are prime ideals, which are maximal ideals? I have no clue about this.

Any suggestion would be appreciated! Thank you!

user26857
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Blackgirl5
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  • By $\mathbb{F}4$, do you mean the (non-field) ring $\mathbb{Z}/4\mathbb{Z}$ or the unique field with 4 elements $\mathbb{F}{2^2}$? In the latter case, its elements are not $0,1,2,3$, rather $0,1,\alpha,\alpha+1$ where $\alpha$ is root of $X^2+X+1$. – Angelo Rendina Nov 06 '16 at 16:06
  • I think it is the first case. – Blackgirl5 Nov 06 '16 at 16:12
  • I actually think it is the latter case, since someone else asked the same question and they also said $\mathbb{F}4$ rather than $\mathbb{Z}_4$ (which is synonim for $\mathbb{Z}/4\mathbb{Z}$). Also, the FIELD $\mathbb{F}{2^2}$ is easier to study because you know its ideals, right? (hint: they are 2) – Angelo Rendina Nov 06 '16 at 16:17
  • I think I got it, thank you! – Blackgirl5 Nov 06 '16 at 16:23

1 Answers1

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The ideals of $\mathbb{Z}$ are exactly $(n)$ for $n\in\mathbb{Z}$. The ideals of $\mathbb{F}_4$ are $(0)$ and $(1)=\mathbb{F}_4$ itself, in particular they are both principal. Can you show now that every ideal of $\mathbb{Z}\times\mathbb{F}_4$ is generated by a single element $(a,b)$ i.e. every ideal is principal?

Now, consider the quotients $\mathbb{Z}\times\mathbb{F}_4/(n)\times(0)$ and $\mathbb{Z}\times\mathbb{F}_4/(n)\times(1)$ for any $n\in\mathbb{Z}$. In the first case you get (isomorphic copy of) $\mathbb{Z}_n\times\mathbb{F}_4$, in the second case $\mathbb{Z}_n\times {0}\equiv\mathbb{Z}_n$. Remember the characterization of prime/maximal ideals in terms of quotients by them...

Angelo Rendina
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