# Artinian module

The term Artinian ring or Artinian module (after Emil Artin ) describes a certain finiteness condition in the mathematical sub-area of algebra . The term has some analogies to the term Noetherian ring , but the two terms are not linked in a very simple way. For example, every Artinian ring is noetherian, but not the other way around.

## Artinian module

### definition

A module over a ring with is called Artinian if it fulfills one of the following equivalent conditions: ${\ displaystyle M}$ ${\ displaystyle R}$ ${\ displaystyle 1}$ • Every non-empty set of -sub-modules of has a minimal element regarding inclusion.${\ displaystyle R}$ ${\ displaystyle M}$ • Each descending sequence of sub-modules becomes stationary, i.e. H. in a chain
${\ displaystyle M_ {1} \ supseteq M_ {2} \ supseteq M_ {3} \ supseteq \ dotsb}$ there is an index , such that for all the following applies: .${\ displaystyle n}$ ${\ displaystyle i> n}$ ${\ displaystyle M_ {i} = M_ {n}}$ • For every family of sub-modules there is a finite subset of such that:${\ displaystyle \ left (M_ {i} \ right) _ {i \ in I}}$ ${\ displaystyle I_ {0}}$ ${\ displaystyle I}$ ${\ displaystyle \ bigcap _ {i \ in I} M_ {i} = \ bigcap _ {i \ in I_ {0}} M_ {i}}$ ### Examples

• Every finite module is Artinian.
• Every finitely generated module over an Artinian ring is Artinian.
• ${\ displaystyle \ mathbb {Z}}$ is not an Artinian module.${\ displaystyle \ mathbb {Z}}$ • A finite direct sum of Artinian modules is Artinian.
• If an (associative) algebra is over a field and has a module of finite dimension, then is Artinian . For example, the rings and are Artinian.${\ displaystyle R}$ ${\ displaystyle K}$ ${\ displaystyle R}$ ${\ displaystyle M}$ ${\ displaystyle K}$ ${\ displaystyle M}$ ${\ displaystyle K \ times K}$ ${\ displaystyle K [T] / (T ^ {n})}$ • The examiner group as a module is Artinian, but not .${\ displaystyle \ mathbb {Z} \ left [{\ tfrac {1} {p}} \ right] {\ Big /} \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Z} \ left [{\ tfrac {1} {p}} \ right]}$ ### properties

• Every injective endomorphism is an automorphism
• For an exact sequence of modules are equivalent:${\ displaystyle 0 \ rightarrow M_ {1} \ rightarrow M_ {2} \ rightarrow M_ {3} \ rightarrow 0}$ 1. ${\ displaystyle M_ {2}}$ is artinsch
2. ${\ displaystyle M_ {1}, M_ {3}}$ are artinsch
• For a (left) module over a (left) Artinian ring are equivalent: ${\ displaystyle M}$ ${\ displaystyle R}$ • M is (left) Artinian
• M is (left) noetherian
• M is finitely generated

## Artinian ring

### definition

A ring is called linksartinsch if artinsch is as -link module. ${\ displaystyle R}$ ${\ displaystyle R}$ ${\ displaystyle R}$ A ring is called rechtsartinsch if artinsch is a right module. ${\ displaystyle R}$ ${\ displaystyle R}$ ${\ displaystyle R}$ A ring is called Artinian when left and right are Arctic. ${\ displaystyle R}$ ${\ displaystyle R}$ (Note: The sub-modules are then precisely the (left / right) ideals .)

### Examples

• Bodies are artinian
• Let be a field, a finitely generated -algebra (i.e. for a suitable ideal ), then an Artinian ring is if and only if .${\ displaystyle K}$ ${\ displaystyle R}$ ${\ displaystyle K}$ ${\ displaystyle R \ simeq K [X] / I}$ ${\ displaystyle I \ subseteq K [X]}$ ${\ displaystyle R}$ ${\ displaystyle dim_ {K} (R) <\ infty}$ • ${\ displaystyle {\ begin {pmatrix} \ mathbb {Z} & \ mathbb {Q} \\ 0 & \ mathbb {Q} \ end {pmatrix}}}$ is right-noetherian, but neither leftartinian nor left-noetherian.
• ${\ displaystyle {\ begin {pmatrix} \ mathbb {Q} & \ mathbb {R} \\ 0 & \ mathbb {R} \ end {pmatrix}}}$ is right-aristocratic but not left-aristocratic.