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Consider the ring generated by $a,b,c,d,e$ subject to the relation $a=bc+de$ and all its cyclic shifts: $b=cd+ea$, and 3 more. It is "idempotent" obviously. Can it be killed by one relation? I will ask Agata Smoktunowicz. She should be able to figure it out quickly.

Update Agata responded saying that the problem, while interesting, is too difficult. She did try using Groebner-Shirshov bases but without success. She did manage to prove the statement for semigroup algebras using an argument similar to Ozawa's (as Ben Steinberg asked here). If $S$ is a semigroup, $S^2=S$, then $KS$ is generated as an ideal by one element.

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Consider the ring generated by $a,b,c,d,e$ subject to the relation $a=bc+de$ and all its cyclic shifts: $b=cd+ea$, and 3 more. It is "idempotent" obviously. Can it be killed by one relation? I will ask Agata Smoktunowicz. She should be able to figure it out quickly.

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Consider the ring generated by $a,b,c,d,e$ subject to the relation $a=bc+de$ and all its cyclic shifts: $b=cd+ea$, and 3 more. It is "idempotent" obviously. Can it be killed by one relation?