Definition:Primary Ideal/Definition 1

Definition
Let $R$ be a commutative ring.

A proper ideal $\mathfrak q$ of $R$ is called a primary ideal :
 * $ x y \in \mathfrak q \implies x \in \mathfrak q \; \vee \; \exists n \in \Z_{>0} : y^n \in \mathfrak q$

Also see

 * Equivalence of Definitions of Primary Ideal of Commutative Ring