artinian
A module $M$ is artinian^{} if it satisfies the following equivalent^{} conditions:

•
the descending chain condition^{} holds for submodules^{} of $M$;

•
every nonempty family of submodules of $M$ has a minimal element.
A ring $R$ is left artinian if it is artinian as a left module over itself (i.e. if ${}_{R}R$ is an artinian module), and right artinian if it is artinian as a right module over itself (i.e. if ${R}_{R}$ is an artinian module), and simply artinian if both conditions hold.
Title  artinian 

Canonical name  Artinian 
Date of creation  20130322 12:26:46 
Last modified on  20130322 12:26:46 
Owner  antizeus (11) 
Last modified by  antizeus (11) 
Numerical id  6 
Author  antizeus (11) 
Entry type  Definition 
Classification  msc 16D10 
Synonym  left artinian 
Synonym  right artinian 
Related topic  Noetherian^{} 
Related topic  Noetherian2 
Related topic  HollowMatrixRings 