noetherian ring

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

↦

x

⁢

r

). 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

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

),

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_Noether

Wikipedia 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

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