Prime element - Knowlpedia

Let $R$ be an integral domain. An element $p\in R$ is a prime element if $p\neq 0$ , $p$ is not a unit, and whenever $p$ divides a product $ab$ , then $p$ divides $a$ or $p$ divides $b$ .

Remarks

Prime elements correspond to prime ideals: $p$ is prime if and only if the principal ideal $(p)$ is a prime ideal. Prime elements are always irreducible, and in a UFD the converse holds.

Examples

In $\mathbb{Z}$ , an integer is prime (in the usual number-theoretic sense) exactly when it is a prime element.
In $k[x]$ for a field $k$ , the polynomial $x$ is a prime element.
In $\mathbb{Z}$ , the element $6$ is not prime since $6\mid 2\cdot 3$ but $6\nmid 2$ and $6\nmid 3$ .