Let $R \neq 0$ be a commutative ring. I think we have

$$ R \text{ is a domain } \iff (0) \text{ is a prime ideal of } R.$$

The *argument* is straightforward: let $a,b \in R$ such that $a\cdot b = 0$. Then to say that $(0)$ is prime is equivalent to say that at least one of $a$ or $b$ is zero, i.e. that $R$ is an integral domain.

Do you agree with this or am I missing some possible issues?