Irreducible element

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.

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.