请输入您要查询的字词:

 

单词 NoetherianRing
释义

noetherian ring


A ring R is right noetherianPlanetmathPlanetmath if it is a right noetherian module (http://planetmath.org/NoetherianModule), considered as a right module over itself in the natural way (that is, an element r acts by xxr). Similarly, R is left noetherian if it is a left noetherian module over itself (equivalently, if the opposite ring of R is right noetherian). We say that R is noetherian if it is both left noetherian and right noetherian.

Examining the definition, it is relatively easy to show that R is right noetherian if and only if the three equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath conditions hold:

  1. 1.

    right idealsMathworldPlanetmathPlanetmath are finitely generatedMathworldPlanetmathPlanetmath,

  2. 2.

    the ascending chain conditionMathworldPlanetmathPlanetmathPlanetmath holds on right ideals, or

  3. 3.

    every nonempty family of right ideals has a maximal elementMathworldPlanetmath.

Examples of Noetherian rings include:

  • any field (as the only ideals are 0 and the whole ring),

  • the ring of integers (each ideal is generated by a single integer, the greatest common divisorMathworldPlanetmathPlanetmath of the elements of the ideal),

  • the p-adic integers (http://planetmath.org/PAdicIntegers), p for any prime p, where every ideal is generated by a multipleMathworldPlanetmathPlanetmath of p, and

  • the ring of complex polynomials in two variables, where some ideals (the ideal generated byPlanetmathPlanetmath X and Y, for example) are not principal, but all ideals are finitely generated.

The Hilbert Basis TheoremMathworldPlanetmath says that a ring R is noetherian if and only if the polynomial ring R[x] is.

A ring can be right noetherian but not left noetherian.

The word noetherian is used in a number of other places. A topology can be noetherian (http://planetmath.org/NoetherianTopologicalSpace); although this is not related in a simple way to the property for rings, the definition is based on an ascending chain condition. A site can also be noetherian; this is a generalizationPlanetmathPlanetmath of the notion of noetherian for topological space.

Noetherian rings (and by extensionPlanetmathPlanetmathPlanetmath most other uses of the word noetherian) are named after Emmy Noether (see http://en.wikipedia.org/wiki/Emmy_NoetherWikipedia for a short biography) who made many contributions to algebraPlanetmathPlanetmath. Older references tend to capitalize the word (Noetherian) but in some fields, such as algebraic geometryMathworldPlanetmathPlanetmath, the word has come into such common use that the capitalization is dropped (noetherian). A few other objects with proper names have undergone this process (abelianMathworldPlanetmath, for example) while others have not (Galois groups, for example). Any particular work should of course choose one convention and use it consistently.

Titlenoetherian ring
Canonical nameNoetherianRing
Date of creation2013-03-22 11:44:52
Last modified on2013-03-22 11:44:52
Ownerarchibal (4430)
Last modified byarchibal (4430)
Numerical id18
Authorarchibal (4430)
Entry typeDefinition
Classificationmsc 16P40
Classificationmsc 18-00
Classificationmsc 18E05
Synonymnoetherian
Related topicArtinianPlanetmathPlanetmath
Related topicNoetherianModule
Related topicNoetherian2
Definesleft noetherian
Definesright noetherian
Definesleft noetherian ring
Definesright noetherian ring
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 4:56:02