irreducible element

In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.

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 $a \mid bc$ for some $b$ and $c$ in $R,$ then $a \mid b$ or $a \mid c.$) In an integral domain, every prime element is irreducible,Sharpe (1987) p.54 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 as noted above $x$ is prime, and so the ideal generated by $x$ is a prime (hence irreducible) ideal of $D$.

Example

In the quadratic integer ring $\mathbf$sqrt 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,
:$3 \mid \left(2 + \sqrt\right)\left(2 - \sqrt\right)=9,$

but 3 does not divide either of the two factors.William W. Adams and Larry Joel Goldstein (1976), ''Introduction to Number Theory'', p. 250, Prentice-Hall, Inc.,

See also

* Irreducible polynomial

References

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{{DEFAULTSORT:Irreducible Element
Ring theory
Algebraic properties of elements