# Difference between revisions of "Noetherian"

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We say that a ring <math>R</math> is left (right) noetherian if it is noetherian as a left (right) <math>R</math>-module. If <math>R</math> is both left and right noetherian, we call it simply noetherian. | We say that a ring <math>R</math> is left (right) noetherian if it is noetherian as a left (right) <math>R</math>-module. If <math>R</math> is both left and right noetherian, we call it simply noetherian. | ||

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## Revision as of 18:03, 9 September 2008

Let be a ring and a left -module. Then we say that is a **noetherian module** if it satisfies the following property, known as the ascending chain condition (ACC):

- For any ascending chain of submodules of , there exists an integer so that (i.e. the chain eventually terminates).

**Theorem.** The following conditions are equivalent for a left -module:

- is noetherian.
- Every submodule of is finitely generated (i.e. can be written as for some ).
- For every collection of submodules of , there is a maximal element.

(The second condition is also frequently used as the definition for noetherian.)

We also have right noetherian modules with the appropriate adjustments.

We say that a ring is left (right) noetherian if it is noetherian as a left (right) -module. If is both left and right noetherian, we call it simply noetherian.

*This article is a stub. Help us out by expanding it.*