# Irreducible element

In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.

## Relationship with prime elements

Irreducible elements should not be confused with prime elements. (A non-zero non-unit element ${\displaystyle a}$ in a commutative ring ${\displaystyle R}$ is called prime if, whenever ${\displaystyle a|bc}$ for some ${\displaystyle b}$ and ${\displaystyle c}$ in ${\displaystyle R}$, then ${\displaystyle a|b}$ or ${\displaystyle a|c}$.) In an integral domain, every prime element is irreducible,[1][2] but the converse is not true in general. The converse is true for UFDs[2] (or, more generally, GCD domains.)

Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal. However, if ${\displaystyle D}$ is a GCD domain, and ${\displaystyle x}$ is an irreducible element of ${\displaystyle D}$, then the ideal generated by ${\displaystyle x}$ is a prime ideal of ${\displaystyle D}$.[3]

## Example

In the quadratic integer ring ${\displaystyle \mathbf {Z} [{\sqrt {-5}}]}$, it can be shown using norm arguments that the number 3 is irreducible. However, it is not a prime element in this ring since, for example,

${\displaystyle 3|\left(2+{\sqrt {-5}}\right)\left(2-{\sqrt {-5}}\right)=9}$

but ${\displaystyle 3}$ does not divide either of the two factors.[4]