noetherian

A module $M$ is noetherian if it satisfies the following equivalent conditions:

• the ascending chain condition holds for submodules of $M$ ;

• every nonempty family of submodules of $M$ has a maximal element;

• every submodule of $M$ is finitely generated.

A ring $R$ is left noetherian if it is noetherian as a left module over itself (i.e. if ${}_{R}R$ is a ), and right noetherian if it is noetherian as a right module over itself (i.e. if $R_R$ is an ), and simply noetherian if both conditions hold.

Title	noetherian
Canonical name	Noetherian
Date of creation	2013-03-22 12:26:53
Last modified on	2013-03-22 12:26:53
Owner	antizeus (11)
Last modified by	antizeus (11)
Numerical id	5
Author	antizeus (11)
Entry type	Definition
Classification	msc 16P40
Synonym	left noetherian
Synonym	right noetherian
Related topic	Artinian
Related topic	Noetherian
Related topic	HollowMatrixRings