请输入您要查询的字词:

 

单词 ImageIdealOfDivisor
释义

image ideal of divisor


Theorem.  If an integral domainMathworldPlanetmath 𝒪 has a divisor theory  𝒪*𝔇,  then the subset [𝔞] of 𝒪, consisting of 0 and all elements divisible by a divisorMathworldPlanetmathPlanetmath 𝔞, is an ideal of 𝒪.  The mapping

𝔞[𝔞]

from the set 𝔇 of divisors into the set of ideals of 𝒪 is injective and maps any principal divisor (α) to the principal idealMathworldPlanetmath (α).

Proof.  Let  α,β[𝔞]  and  λ𝒪.  Then, by the postulate 2 of divisor theory (http://planetmath.org/DivisorTheory), α-β is divisible by 𝔞 or is 0, and in both cases belongs to [𝔞].  When  λα0,  we can write  (α)=𝔞𝔠  with 𝔠 a divisor.  According to the homomorphicity of the mapping  𝒪*𝔇,  we have

(λα)=(λ)(α)=(λ)𝔞𝔠,

and therefore the element λα is divisible by 𝔞, i.e. λα[𝔞].  Thus, [𝔞] is an ideal of 𝒪.

The injectivity of the mapping  𝔞[𝔞]  follows from the postulate 3 of divisor theory (http://planetmath.org/DivisorTheory).

The ideal [𝔞] may be called the image ideal of 𝔞 or the ideal determined by the divisor 𝔞.

Remark.  There are integral domains 𝒪 having a divisor theory but also having ideals which are not of the form [𝔞] (for example a polynomial ring in two indeterminates and its ideal formed by the polynomials without constant term).  Such rings have ‘‘too many ideals’’.  On the other hand, in some integral domains the monoid of principal ideals cannot be embedded into a free monoid; thus those rings cannot have a divisor theory.

References

  • 1 М. М. Постников:Введение  в  теорию  алгебраических чисел.  Издательство ‘‘Наука’’. Москва (1982).
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 20:51:41