# What are the rings in which left and right zero divisors coincide called?

A unital ring $R$ is reversible iff $ab=0\implies ba=0.$ This condition implies the following one.

If $a\in R$ is a left-zero divisor, then $a$ is also a right-zero divisor. And the other way around.

Could you please tell me if such rings have a name? It is a strictly weaker condition because it holds in $M_2(\mathbb R)$, which isn't reversible. Indeed, $$\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\1&1\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix},$$ but $$\begin{pmatrix}0&0\\1&1\end{pmatrix}\begin{pmatrix}1&0\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\1&0\end{pmatrix}.$$

Also, is there any relation between this condition and Dedekind-finiteness? I know that reversibility implies Dedekind finiteness (the proof is here).

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