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  20130322 12:26:53 
Last modified on  20130322 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 