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.

properties

• An Artinian ring is noetherian
• More precisely, a commutative ring with one element is Artinian if and only if it is Noetherian and zero-dimensional (i.e. if every prime ideal is a maximal ideal )
• An Artinian ring of integrity is already a body. The following even stronger statement applies: An integrity ring that fulfills the descending chain condition for main ideals is a body.
• If the null ideal in a ring is the product of maximum ideals , then the ring is Artinian precisely when it is noetherian
• In an Artinian ring, every prime ideal is already maximal
• In an Artinian ring there are only finitely many maximum ideals (and thus only finitely many prime ideals )
• In an Artinian ring, the Nile radical is nilpotent
• Every Artinian ring is a finite product of local Artinian rings

literature

• KA Zhevlakov: Artinian ring . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).