PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (37)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
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$



"primal element" is owned by CWoo.
(view preamble)

View style:

Also defines:  primal
Log in to rate this entry.
(view current ratings)

Cross-references: inverse, unit, prime element, irreducible, commutative ring
There are 3 references to this entry.

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 2276 times total.

Classification:
AMS MSC: 13A05 (Commutative rings and algebras :: General commutative ring theory :: Divisibility)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)