This is a follow-up question to Is there a reduced ring with exactly $3$ idempotents?, to which the answer was "no."
Note: In this question, 'ring' means ring with unity, but not necessarily commutative
In fact, in a (non-trivial) reduced ring, the number of idempotents is either even or $\infty$. The reason is that the idempotents come in pairs $e,1-e$. And $e \neq1-e$, otherwise $ee=e-ee$ and $e^2=0$, implying (since the ring is reduced) that $e=0$, which can't happen if $e=1-e$.
My next question is, does there exist a reduced ring whose number of idempotents is a multiple of $3$? (For example, can we find a reduced ring with $6$ idempotent elements? $12$? $18$? $3000$?)
What about rings in general? (i.e. not necessarily reduced)
Attempting the easiest case first, assume $R$ is a reduced ring and the idempotent elements are $\{0,1,a,(1-a),b,(1-b)\}$ (all distinct). I see that the product of two idempotents must be idempotent (since the idempotents commute with everything). Also, I see that the square of the difference of two idempotents must also be idempotent. So $ab \in \{0,1,a,(1-a),b,(1-b)\}$ . (I suspect that there might be a way to derive a contradiction from this, although I don't see how to do so yet.)
from Hot Weekly Questions - Mathematics Stack Exchange
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