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 $a$ 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, but the converse is not true in general. The converse is true for unique factorization domains (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 a prime ideal of $D$.

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,

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

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

See also

• Irreducible polynomial

References

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