Definition:Primary Ideal/Definition 1

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

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

Also see

 * Equivalence of Definitions of Primary Ideal of Commutative Ring