Ring with Unity has Prime Ideal

Theorem
Let $A$ be a non-trivial commutative ring with unity.

Then $A$ has a prime ideal.

Proof
By Krull's Theorem, $A$ has a maximal ideal.

By Maximal Ideal of Commutative and Unitary Ring is Prime Ideal, $A$ has a prime ideal.

Also see

 * Spectrum of Ring is Nonempty