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.