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primal element (Definition)

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$.

Remark. 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$. $ \qedsymbol$



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Cross-references: inverse, unit, prime element, irreducible, commutative ring
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This is version 4 of primal element, born on 2004-11-22, modified 2007-05-11.
Object id is 6508, canonical name is PrimalElement.
Accessed 2361 times total.

Classification:
AMS MSC: 13A05 (Commutative rings and algebras :: General commutative ring theory :: Divisibility)
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