primal element

An element r in a commutative ring R is called primal if whenever rab, with a,bR, then there exist elements s,tR such that

  1. 1.


  2. 2.

    sa and tb.

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


Suppose a is irreducible and primal, and abc. Since a is primal, there is x,yR such that a=xy, with xb and yc. Since a is irreducible, either x or y is a unit. If x is a unit, with z as its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath, then za=zxy=y, so that ay. But yc, we have that ac. ∎

Title primal element
Canonical name PrimalElement
Date of creation 2013-03-22 14:50:21
Last modified on 2013-03-22 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