# 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.

Irreducible elements should not be confused with prime elements. (A non-zero non-unit element $a$ in a commutative ring $R$ is called prime if, whenever $a | bc$ for some $b$ and $c$ in $R$, then $a|b$ or $a|c$.) 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 $D$ is a GCD domain, and $x$ is an irreducible element of $D$, then the ideal generated by $x$ is an irreducible ideal of $D$.[3]

## Example

In the quadratic integer ring $\mathbf{Z}[\sqrt{-5}]$, 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,

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

but $3$ does not divide either of the two factors.[4]

## References

1. ^ Consider p a prime that is reducible: p=ab. Then p | ab => p | a or p | b. Say p | a => a = pc, then we have: p=ab=pcb => p(1-cb)=0. Because R is an integral domain we have: cb=1. So b is a unit and p is irreducible.
2. ^ a b Sharpe (1987) p.54
3. ^ http://planetmath.org/encyclopedia/IrreducibleIdeal.html
4. ^ William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9