请输入您要查询的字词:

 

单词 PolynomialRingWhichIsPID
释义

polynomial ring which is PID


Theorem.  If a polynomial ring D[X] over an integral domainMathworldPlanetmath D is a principal ideal domainMathworldPlanetmath, then coefficient ring D is a field. (Cf. the corollary 4 in the entry polynomial ring over a field.)

Proof.  Let a be any non-zero element of D.  Then the ideal  (a,X)  of D[X] is a principal idealMathworldPlanetmathPlanetmath (f(X)) with f(X) a non-zero polynomial (http://planetmath.org/ZeroPolynomial2).  Therefore,

a=f(X)g(X),X=f(X)h(X)

with g(X) and h(X) certain polynomials in D[X].  From these equations one infers that f(X) is a polynomial c and h(X) is a first degree polynomial b0+b1X (b10).  Thus we obtain the equation

cb0+cb1X=X,

which shows that cb1 is the unity 1 of D.  Thus  c=f(X)  is a unit of D, whence

(a,X)=(f(X))=(1)=D[X].

So we can write

1=au(X)+Xv(X),

where  u(X),v(X)D[X].  This equation cannot be possible without that a times the constant term of u(X) is the unity.  Accordingly, a has a multiplicative inverseMathworldPlanetmath in D.  Because a was arbitrary non-zero elenent of the integral domain D, D is a field.

References

  • 1 David M. Burton: A first course in rings and ideals. Addison-Wesley Publishing Company. Reading, Menlo Park, London, Don Mills (1970).
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 13:10:33