Prime Ideal is Primary Ideal

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

Let $\mathfrak p$ be a prime ideal of $R$.

Then $\mathfrak p$ is a primary ideal of $R$.

Proof
Let $xy \in \mathfrak p$.

Let $x \not \in \mathfrak p$

By definition of prime ideal:
 * $y^1 = y \in \mathfrak p$

Thus, by definition, $\mathfrak p$ is a primary ideal.