Category:Primary Ideals

This category contains results about Primary Ideals.
Definitions specific to this category can be found in Definitions/Primary Ideals.

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.