Q:

Ideals of $\mathbb Z\times\mathbb F_4$

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!

A:

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...

