Let $R$ be a reduced ring (let's assume with identity, although not necessarily commutative).
Is it possible for $R$ to have exactly 3 idempotent elements? If so, what would be an example?
I know that the idempotents of a reduced ring commute with everything in the ring. This implies that the product of two idempotents is idempotent. But, alas, I don't see how this would help. (After all, if $0$ and $1$ and $u$ are the only idempotents, we can take products of these elements but we don't get anything new.)
Edit: I see now that it is impossible for $R$ to have exactly $3$ idempotents. (Thank you to the commentors for your insight!) I am still, however, interesting in doing further investigation on the structure of reduced rings. I have now posted a more general follow-up question Can a reduced ring have (# idempotents) $\in 3 \mathbb{Z}$?
No.
Assume $\{0,1,e\}$ are the only idempotents. (All distinct).
By Bernard's comment, $1-e$ is idempotent. We can't have $1-e=0$, otherwise $e=1$. We can't have $1-e=1$, otherwise $e=0$. So we must have $e=1-e$. So $e^2=e-e^2=0$. Since $e^2=0$ and our ring is a reduced ring, we must have $e=0$, a contradiction.