请输入您要查询的字词:

 

单词 IrreducibleOfAUFDIsPrime
释义

irreducible of a UFD is prime


Any irreducible elementMathworldPlanetmath of a factorial ring D is a prime elementMathworldPlanetmath of D.

Proof.  Let p be an arbitrary irreducible element of D.  Thus p is a non-unit.  If ab(p){0},  then  ab=cp  with cD.  We write a,b,c as products of irreducibles:

a=p1pl,b=q1qm,c=r1rn

Here, one of those first two products may me empty, i.e. it may be a unit.  We have

p1plq1qm=r1rnp.(1)

Due to the uniqueness of prime factorizationMathworldPlanetmath, every factor rk is an associateMathworldPlanetmath of certain of the l+m irreducibles on the left hand side of (1).  Accordingly, p has to be an associate of one of the pi’s or qj’s.  It means that either  a(p)  or  b(p).  Thus, (p) is a prime idealMathworldPlanetmathPlanetmath of D, and its generatorPlanetmathPlanetmathPlanetmath must be a prime element.

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 3:39:38