Maximal and Prime ideals of cartesian product of rings

I want to find the maximal and prime ideals of $\mathbb{Z}_4 \times \mathbb{Z_4}$ and $\mathbb{Z}_5 \times \mathbb{Z}_6$.

I figured out in a previous post that the ideals of $\mathbb{Z}/4 \times \mathbb{Z}/4$ are a total of $9$.

I know that a maximal ideal is a proper ideal which cannot be contained in a larger ideal. I've been able to find these for single rings like $\mathbb{Z_12}$, but I am unsure how they work for cross products.

If $M_1$ and $M_2$ are the maximal ideals of $R_1; R_2$ where $R_1,R_2$ are rings, then are the maximal ideals of $R_1 \times R_2$ are $M_1 \times M_2$?

Solution 1:

The maximal ideals of a product are $M_1\times R_2$ and $R_1\times M_2$. $M_1\times M_2$ is never a maximal ideal because the two proper ideals above contain it.