# Category:Primary Ideals

Jump to navigation
Jump to search

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.

## Pages in category "Primary Ideals"

The following 5 pages are in this category, out of 5 total.