# 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 unique factorization domains[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 as noted above ${\displaystyle x}$  is prime and so 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,

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

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

## References

1. ^ Consider ${\displaystyle p}$  a prime element of ${\displaystyle R}$  and suppose ${\displaystyle p=ab.}$  Then ${\displaystyle p|ab\Rightarrow p|a}$  or ${\displaystyle p|b.}$  Say ${\displaystyle p|a\Rightarrow a=pc,}$  then we have ${\displaystyle p=ab=pcb\Rightarrow p(1-cb)=0.}$  Because ${\displaystyle R}$  is an integral domain we have ${\displaystyle cb=1.}$  So ${\displaystyle b}$  is a unit and ${\displaystyle p}$  is irreducible.
2. ^ a b Sharpe (1987) p.54
3. ^ "Archived copy". Archived from the original on 2010-06-20. Retrieved 2009-03-18.CS1 maint: Archived copy as title (link)
4. ^ William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9