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Let R be an associative ring of unity. An element $a$ in $R$ is called a commutator if $a$ can be expressed as $a=xy-yx$ for some $x,y\in R$. If $R$ is a division ring, that is a ring whose nonzero elements are invertible, then every commutator in $R$ is not a zero-divisor. I want to establish a condition to be that all commutators in $R$ are not zero-divisors.

Any counterexample or reference or technique is very much appreciated. I appreciate any help you can provide.

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    $\begingroup$ "then every commutator in $R$ is not a zero-divisor." To be clear, your definition of zero divisor excludes zero itself? Iff that is the case, why would you start at division rings instead of skipping straight to domains? (Where there are simply no nonzero-zero-divisors $\endgroup$
    – rschwieb
    May 25, 2023 at 14:10
  • $\begingroup$ Yes, of course, domains are a good example. I want to establish a necessary and sufficient condition to be that all commutators are not zero-divisors. $\endgroup$ May 25, 2023 at 14:47

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Let $R=M_2(K)$ with $K$ field, and let $e=\big(\begin{smallmatrix} 1 & 0\\ 0 & 0 \end{smallmatrix}\big)$ and $f=\big(\begin{smallmatrix} 0 & 1\\ 0 & 0 \end{smallmatrix}\big)$. Then we have $f=ef-fe$ and $f^2=0$, and so $f$ is a commutator and a zero divisor at the same time.

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    $\begingroup$ @rschwieb Yes, true. I have edited it, I think it is more clear now. $\endgroup$ May 25, 2023 at 14:22
  • $\begingroup$ Makes sense now. thanks! $\endgroup$
    – rschwieb
    May 25, 2023 at 15:40

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