Is it true that in simple rings every regular element is invertible?

Question

$r$ is regular element of ring $R$ if $rx=0$ implies $x=0$ or $xr=0$ implies $x=0$. I think this is true in matrix rings over divition ring or maybe in all artinian simple rings.

Answer 1

No. There exists a simple domain that is not a division ring, namely they first Weyl algebra.

All nonzero elements are regular, but not all of them are invertible.