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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. –  Pound Dec 24 '11 at 2:08

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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? –  John E. Smith Dec 24 '11 at 0:26
    
Sure. But there is a reason why people love quivers.... :D –  Mariano Suárez-Alvarez Dec 24 '11 at 0:28
    
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. –  John E. Smith Dec 24 '11 at 0:30
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Could you recommend a good introduction to quivers? –  John E. Smith Dec 24 '11 at 0:43
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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. –  Mariano Suárez-Alvarez Dec 24 '11 at 0:48

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