# Prime element

From Commalg

Template:Associate-invariant integral domain-element property

## Contents

## Definition

### Symbol-free definition

A nonzero element in an integral domain is said to be a **prime element** if it satisfies the following equivalent conditions:

- Whenever it divides the product of two elements, it must divide at least one of them.
- The principal ideal generated by it is a prime ideal, or equivalently, the quotient ring by this principal ideal is an integral domain.

### Definition with symbols

A nonzero element in an integral domain is said to be *prime'* if it satisfies the following:

- Whenever , then or .
- The principal ideal is a prime ideal in , or equivalently, is an integral domain.

### Invariance up to associates

`Further information: Prime element property is invariant upto associates`

Given two associate elements, one of them is prime if and only if the other one is.