Definition:Primary Ideal/Definition 2

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:

each zero-divisor of the quotient ring $R / \mathfrak q$ is nilpotent.

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