noetherian ring

A ring $R$ is right noetherian 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 $x\mapsto xr$). 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 equivalent conditions hold:

1. right ideals are finitely generated,

2. the ascending chain condition holds on right ideals, or

3. every nonempty family of right ideals has a maximal element.

Examples of Noetherian rings include:

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

• the ring $\mathbb{Z}$ of integers (each ideal is generated by a single integer, the greatest common divisor of the elements of the ideal),

• the $p$-adic integers (http://planetmath.org/PAdicIntegers), ${\mathbb{Z}}_p$ for any prime $p$, where every ideal is generated by a multiple of $p$, and

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

The Hilbert Basis Theorem 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 generalization of the notion of noetherian for topological space.

Noetherian rings (and by extension 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 algebra. Older references tend to capitalize the word (Noetherian) but in some fields, such as algebraic geometry, 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 (abelian, 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	Artinian
Related topic	NoetherianModule
Related topic	Noetherian2
Defines	left noetherian
Defines	right noetherian
Defines	left noetherian ring
Defines	right noetherian ring