Prime element

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 $b$.

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 $6\nmid 3$.