primal element
An element $r$ in a commutative ring $R$ is called primal if whenever $r\mid ab$, with $a,b\in R$, then there exist 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 $a\mid c$. ∎
Title  primal element 

Canonical name  PrimalElement 
Date of creation  20130322 14:50:21 
Last modified on  20130322 14:50:21 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 13A05 
Defines  primal 