Definition:Primary Ideal

Definition

Let $R$ be a commutative ring with unity.

Definition 1

A proper ideal $\mathfrak q$ of $R$ is called a primary ideal if and only if:

$\forall x,y \in R :$

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

Definition 2

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 known as

A primary ideal $\mathfrak q$ is also called $\mathfrak p$-primary, where

$\mathfrak p := \map \Rad {\mathfrak q}$

is the radical of $\mathfrak q$ is the smallest prime ideal including $\mathfrak q$.

See Radical of Primary Ideal is Smallest Prime Ideal.

Also see

• Equivalence of Definitions of Primary Ideal of Commutative Ring
• Prime Ideal is Primary Ideal
• Radical of Primary Ideal is Smallest Prime Ideal

• Results about primary ideals can be found here.