Irreducible element - Imported Knowlpedia Content

Let $R$ be an integral domain. An element $a\in R$ is irreducible if $a\neq 0$ , $a$ is not a unit, and whenever $a=bc$ , then $b$ or $c$ is a unit.

Remarks

Irreducibles are the basic “atoms” of factorization. In general domains, irreducible need not imply prime, but they coincide in a UFD.

Examples

In $\mathbb{Z}$ , $2$ is irreducible.
In $k[x,y]$ (for a field $k$ ), the polynomial $x$ is irreducible.
In $\mathbb{Z}$ , $6$ is not irreducible since $6=2\cdot 3$ with neither factor a unit.