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	2013-03-22 12:26:46
Last modified on	2013-03-22 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