Artinian
Adjective
Meaning
(algebra, of a ring) in which any descending chain of ideals eventually starts repeating.
(algebra, of a module) in which any descending chain of submodules eventually starts repeating.
Source: en.wiktionary.orgPhrases with the word
Anagrams
Examples
Another consequence is that a left Artinian ring is right Noetherian if and only if right Artinian.
If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.
Conversely, every Artinian integral domain is a field.
The integers, however, form a Noetherian ring which is not Artinian.
It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem ).