Idempotents in a ring without unity (rng) and no zero divisors.

abstract-algebra ring-theory idempotents rngs

Solution 1:

Proposition: If a rng $R$ which does not have nonzero zero divisors, a nonzero idempotent of $R$ must be an identity for the ring.

Proof: Let $e$ be a nonzero idempotent. Since $e(er-r)=0=(re-r)e$ for all $r\in R$ and $e$ is nonzero, we conclude $er-r=0=re-r$, and so $e$ is an identity element.

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