Finding the Ideals of a Direct Product of Rings

Question

Let $R_1 = \mathbb{Z}$, and $R_2 = \mathbb{F}_4$. Find all ideals in $R_1\times R_2$. Which of them are principal ideals? Which are prime? Which are maximal?

I know the ideals in $\mathbb{Z}$ and $\mathbb{F}_4$, but I am having trouble finding the ideals in the cartesian product of both of these rings. If someone could help me out it would be appreciated.

Answer 1

For two unitary rings $A,B$ one has that every ideal $I$ of $A\times B$ is actually equal to $J_A\times J_B$, for some ideal $J_A,J_B$ in the respective rings (see Structure of ideals in the product of two rings).

This is the case in your example, so all you need to know is what the ideals of $\mathbb{Z}$ and $\mathbb{F}_4$ are (and then study their properties - maximality, primality, principality).