Does $\mathbb Z/{2}\times\mathbb Z/{2}$ have no maximal and prime ideal?

Question

Since Maximal ideal needs different from Ring,but and other proper ideal that contain it. But I did not found in this case. For Prime Ideal I must not equal Ring. and need $ab\in Ideal\rightarrow a\in Ideal$ or $ b\in Ideal $ I did not found too Does it both exist ?

Answer 1

$\mathbb Z_2 \times 0$ and $0 \times \mathbb Z_2 $ are two prime ideals of $\mathbb Z_2 \times \mathbb Z_2$. They are actually maximal because the quotient is isomorphic to $\mathbb Z_2$, a field.

This happens in every finite commutative ring: every prime ideal is maximal, because every finite integral domain is a field.