Integral Domain iff Zero Ideal is Prime

Theorem
Let $A$ be a commutative ring with unity.


 * $(1): \quad A$ is an integral domain
 * $(2): \quad$ the zero ideal $0 \subseteq A$ is prime

Proof
By Prime Ideal iff Quotient Ring is Integral Domain, $0$ is prime the quotient ring $A / 0$ is an integral domain.

By Quotient Ring by Null Ideal, $A \cong A / 0$.