primal element

An element $r$ in a commutative ring $R$ is called primal if whenever $r\mid ab$, with $a,b\in R$, then thereexist elements $s,t\in R$ such that

1. $r=st$,

2. $s\mid a$ and $t\mid b$.

Lemma. In a commutative ring, an element that is both irreducible and primal is a prime element.

Proof.

Suppose $a$ is irreducible and primal, and $a\mid bc$. Since $a$ is primal, there is $x,y\in R$ such that $a=xy$, with $x\mid b$ and $y\mid c$. Since $a$ is irreducible, either $x$ or $y$ is a unit. If $x$ is a unit, with $z$ as its inverse, then $za=zxy=y$, so that $a\mid y$. But $y\mid c$, we have that $a\mid c$.∎