I was studying about integral domain and wondered that why it should be a commutative ring with unity; that is, I was wondering if there is a noncommutative ring with no zero divisors (1) with/ or (2) without unity, or a (3) commutative ring with no zero divisors with unity.
and I guess this page provided a partial answer; it proved that if a ring $R$ has no zero divisors, a nonzero idempotent($a$ is an idempotent if $a^2=a$) of $R$ must be the unity of $R$. so if I can prove that a ring $R$ with no zero divisors has a nonzero idempotent, then every ring with no zero divisors must have a unity.
every ring has an idempotent since it contains $0$. but (4) is there a nonzero one if the ring has no zero divisors?
I guess the commutativity is less connected with the condition having no zero divisors since the division ring can be noncommutative and has no zero divisors. so the case (1) is solved and (2), (3) are relying on (4).
hence, what I want to know is just (4) as in the title.