Irreducible element

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 ${\displaystyle a\mid bc}$ for some ${\displaystyle b}$ and ${\displaystyle c}$ in ${\displaystyle R,}$ then ${\displaystyle a\mid b}$ or ${\displaystyle a\mid c.}$) 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 (hence irreducible) ideal of ${\displaystyle D}$.[3]

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]

• Irreducible polynomial

• Sharpe, David (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6. Zbl 0674.13008.