$AB \neq 0$ but $BA=0$ Do there exists to matrices or objects such that $AB \neq 0$ but $BA=0$?
Another way to ask this question is if there exists objects or matrices $A$ and $B$ such that...
$[A,B]=AB$
where $[ \, , \, ]$ is the commutator $[A,B]=AB-BA.$
If such matrices do not exist, what does that imply about the algebra that the elements are in?

What about $$B = \begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix} $$ and $$A = \begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}$$
