noetherian ring

A ring R is right noetherianPlanetmathPlanetmath if it is a right noetherian module (, 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.
  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 (, 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 (; 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 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.

Title noetherian ring
Canonical name NoetherianRing
Date of creation 2013-03-22 11:44:52
Last modified on 2013-03-22 11:44:52
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 18
Author archibal (4430)
Entry type Definition
Classification msc 16P40
Classification msc 18-00
Classification msc 18E05
Synonym noetherian
Related topic ArtinianPlanetmathPlanetmath
Related topic NoetherianModule
Related topic Noetherian2
Defines left noetherian
Defines right noetherian
Defines left noetherian ring
Defines right noetherian ring