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We show that for any two $n\times n$ square-zero matrices $A$ and $B$ over a division ring, if the right column spaces of $AB$ and $BA$ are the same, then the rank of $AB$ is at most $n/4$, and if, in addition, the right null spaces of $AB$ and $BA$ are the same, then the rank of $A+B$ is at most $n/2$. This generalizes some known results.
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