I am studying Ring Theory from Herstein's book "Topics in algebra" and I would like to discuss one moment.
Definition 1. Let $R$ be a ring. The element $a\neq 0\in R$ is called zero-divisor if there exists $b\neq 0\in R$ such that $ab=0_R$.
Definition 2. An integral domain is the commutative ring which has not zero-divisors.
Also it is easy prove the following fact: If $D$ is an integral domain of finite characteristics then characteristics is a prime number.
Can trivial ring be an integral domain? If yes then it's characteristics is not prime which contradicts to the above fact!
But Herstein does not say nothing about triviality of integral domain.
Can anyone clarify this question, please?