Artinian
Encyclopedia
In mathematics, Artinian, named for Emil Artin
, is an adjective that describes objects that satisfy particular cases of the descending chain condition.
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
, is an adjective that describes objects that satisfy particular cases of the descending chain condition.
- A ring is an Artinian ringArtinian ringIn abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
if it satisfies the descending chain condition on (one-sided) ideals - A module is an Artinian moduleArtinian moduleIn abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself...
if it satisfies the descending chain condition on submodules.