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 if and only if:

$\forall x,y \in R :$

$x y \in \mathfrak q \implies x \in \mathfrak q \; \lor \; \exists n \in \N_{>0} : y^n \in \mathfrak q$

Also see

• Equivalence of Definitions of Primary Ideal of Commutative Ring

Sources

• 1969: M.F. Atiyah and I.G. MacDonald: Introduction to Commutative Algebra: Chapter $4$: Primary Decomposition