# Irreducible element

From Commalg

Template:Associate-invariant curing-element property

## Contents

## Definition

### In a commutative unital ring

A nonzero element in a commutative unital ring is said to be **irreducible** if it is neither zero nor a unit, and given any factorization of the element as a product of two elements of the ring, it is associate to one of them.

### In an integral domain

In an integral domain, there are two equivalent formulations. A nonzero element in an integral domain is said to be **irreducible** if it is neither zero nor a unit, and it satisfies the following equivalent conditions:

- Any expression of it as a product of two elements has the property that one of the factors is associate to it.
- Any expression of it as a product of two elements has the property that one of the factors is a unit.

## Relation with other properties

### Stronger properties

- Prime element:
*For proof of the implication, refer Prime element implies irreducible and for proof of its strictness (i.e. the reverse implication being false) refer Irreducible element not implies prime* - Universal side divisor:
*For proof of the implication, refer Universal side divisor implies irreducible and for proof of its strictness (i.e. the reverse implication being false) refer Irreducible not implies universal side divisor*