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Let $R$ be a ring with $1 \neq 0$ that contains noncentral idempotents. If for every noncentral idempotent $e$ of $R$ the corner ring $eRe$ is a division ring and $eR(1-e)Re \neq 0$, is the ring $R$ semiprime?

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    You should try merging your accounts.2011-12-24

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If you modify my example by modding out by all paths of length $3$ you get a counterexample.

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    Is there a way to describe this modified example in terms of matrices?2011-12-24
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    Sure. But there is a reason why people love quivers.... :D2011-12-24
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    Fair enough, but for right now, I understand matrices much better. If you can help with this, I'd really appreciate it. Either way thanks for your quick responses.2011-12-24
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    If you like examples (and you should) you simply *must* become familiar with quivers and relations.2011-12-24
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    Could you recommend a good introduction to quivers?2011-12-24
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    The first volume of the book by Assem, Skowroński and Simson on representation theory. The first couple of chapters, in fact.2011-12-24