Equivalence of Definitions of Primary Ideal

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

Definition 1 implies Definition 2
Let $x + \mathfrak q$ be a zero-divisor of $R / \mathfrak q$.

That is, there is a $y \not \in \mathfrak q$ such that:
 * $\paren {x + \mathfrak q} \paren {y + \mathfrak q} = 0 + \mathfrak q$

Thus:
 * $xy + \mathfrak q = 0 + \mathfrak q$

which means:
 * $xy \in \mathfrak q$

Then, :
 * $\exists n \in \N_{>0} : x^n \in \mathfrak q$

so that:
 * $\paren {x + \mathfrak q}^n = x^n + \mathfrak q = 0 + \mathfrak q$

Therefore $x + \mathfrak q$ is nilpotent.

Definition 2 implies Definition 1
Let $xy \in \mathfrak q$ but $x \not \in \mathfrak q$.

That is:
 * $\paren {x + \mathfrak q} \paren {y + \mathfrak q} = 0 + \mathfrak q$

Then,, $\paren {y + \mathfrak q}$ is nilpotent in $R \ \mathfrak q$.

Thus:

so that:
 * $y^n \in \mathfrak q$