A is a 2x2 matrix, if $\displaystyle A^{2}=0$, then the matrix A may not be a zero matrix, right?

Now if I have $\displaystyle AB=0$, then can I conclude that either $\displaystyle A=0$ or $\displaystyle B=0$?

A and B can both be non-zero also, right?

So then can I say that $\displaystyle AB=0$ implies $\displaystyle A=0$ or $\displaystyle B=0$ is not true in general? But when is it true, when is it not true?

Thank you!!