Example of a ring satisfying "no nonzero nilpotent commutators" but not "if $xy=0$, then $yx=0$ for all $x, y \in R$"

Question

Is there any example of a ring satisfying "no nonzero nilpotent commutators" but not "if $xy=0$, then $yx=0$ for all $x, y \in R$"? Of course, such a ring cannot be commutative, but I cannot find an example for the ring. Thank you in advanced.

Answer 1

If $xy=0$, then

$$[x,y]^2 = (xy - yx)^2 = (yx)^2 = yxyx = 0 $$

It follows that either $yx = 0$ or $[x,y]$ is a nonzero nilpotent.