An example of non commutative ring where $ab=0$ but $ba\neq0$

Question

Give an example of a non commutative ring where $ab=0$ but $ba\neq0$. Other disprove the statement.

I cannot find any way to disprove it. I think such an example exists. I was trying with $M_2\left(\mathbb{Z}\right)$ but could not get any example.

Answer 1

Take $a=\left[\begin{smallmatrix}0&1\\0&1\end{smallmatrix}\right]$ and $b=\left[\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right]$.