# Primary ring

From Commalg

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring

View all properties of commutative unital ringsVIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

The property of being an ideal for which the quotient ring has this property is:primary ideal

## Contents

## Definition

### Symbol-free definition

A commutative unital ring is termed a **primary ring** if it satisfies the following equivalent conditions:

- Whenever the product of two elements in it is zero, either the first element is zero, or the second element is nilpotent
- The zero ideal is a primary ideal
- The ring, as a module over itself, has a unique associated prime.

### Definition with symbols

A commutative unital ring is termed a **primary ring** is whenever in , then either or there exists a such that .