$AB \neq 0$ 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?

Sure - here's an example, with matrices taking entries in the field $\mathbb{Z}_{2}$:
$$
\begin{bmatrix}
1 & 1\\
0 & 0\\
\end{bmatrix}
\begin{bmatrix}
1 & 0\\
1 & 0
\end{bmatrix}
=
\begin{bmatrix}
0 & 0\\
0 & 0
\end{bmatrix},
$$
but
$$
\begin{bmatrix}
1 & 0\\
1 & 0\\
\end{bmatrix}
\begin{bmatrix}
1 & 1\\
0 & 0
\end{bmatrix}
=
\begin{bmatrix}
1 & 1\\
1 & 1
\end{bmatrix}.
$$
