Suppose there are non-trivial idempotents in the ring without unity. Is it right that all of them are zero divisors?
If we're given unitary ring with unity $e$ and $a$ is non-trivial idempotent then $e-a \neq 0$. But $a(e-a) = 0$ so $a$ is zero divisor.
But i'm not sure about the case when ring doesn't have unity.
EDIT In the non-commutative case i wonder whether each non-zero idempotent must be SOME kind of zero divisors, i.e. left or right, or perhaps both.
A partial answer:
If the ring is commutative and finite, then the answer is yes. Take $e\in R$ an idempotent and define $f:R\to R$ by $f(a)=ea$. If $e$ is not a zerodivisor then $f$ is injective and therefore bijective. Now prove that $e$ is the unity of $R$: if $b\in R$ then there exists $a\in R$ such that $b=ea$ and we have $eb=e(ea)=ea=b$.