Integral Domain iff Zero Ideal is Prime

Theorem

Let $A$ be a commutative ring with unity.

The following statements are equivalent:

$(1): \quad A$ is an integral domain

$(2): \quad$ the zero ideal $\ideal 0 \subseteq A$ is prime

Proof

By Prime Ideal iff Quotient Ring is Integral Domain, $\ideal 0$ is prime if and only if the quotient ring $A / \ideal 0$ is an integral domain.

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

$\blacksquare$