Integral Domain iff Zero Ideal is Prime

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Theorem

Let $A$ be a commutative ring with unity.

The following are equivalent:

$(1): \quad A$ 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 if and only if the quotient ring $A / 0$ is an integral domain.

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

$\blacksquare$