请输入您要查询的字词:

 

单词 ExamplesOfRadicalsOfIdealsInCommutativeRings
释义

examples of radicals of ideals in commutative rings


Let R be a commutative ring. Recall, that ideals I,J in R are called coprimeMathworldPlanetmathPlanetmath iff I+J=R. It can be shown, that if I,J are coprime, then IJ=IJ. Elements x1,,xnR are called pairwise coprime iff (xi)+(xj)=R for ij. It follows by induction, that for pairwise coprime x1,,xnR we have (x1xn)=(x1)(xn),

Let xR be such that

x=p1α1pnαn,

for some prime elementsMathworldPlanetmath piR, αi and assume that p1,,pn are coprime. Denote by

x¯=p1pn.

We shall denote by r(I) the radicalPlanetmathPlanetmathPlanetmath of an ideal IR.

Proposition. r((x))=(x¯).

Proof. ,,” Let α=max(α1,,αn). Then we have

x¯α=(p1pn)α=p1αpnα=p1α-α1pnα-αnp1α1pnαn=yx

and thus x¯α(x). This shows the first inclusion.

,,” Assume that yr((x)) and y0. Then there is n such that yn(x). Thus x divides yn. Of course for any i{1,,n} we have that pi divides x. Thus pi divides yn and since pi is prime, we obtain that pi divides y. Now for ij elements pi and pj are coprime, thus x¯ divides y and therefore y(x¯), which completesPlanetmathPlanetmath the proof.

Remark. If we assume that R is a PID (and thus UFD), then the previous proposition gives us the full characterization of radicals of ideals in R. In particular an ideal in PID is radical if and only if it is generated by an element of the form p1pn, where for ij elements pi and pj are not associated primes.

Examples. Consider ring of integersMathworldPlanetmath . Then we have:

r((12))=(6);
r((9))=(3);
r((7))=(7);
r((1125))=(15).
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 7:16:19