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:
- Every non-empty set of -sub-modules of has a minimal element regarding inclusion.
- Each descending sequence of sub-modules becomes stationary, i.e. H. in a chain
- there is an index , such that for all the following applies: .
- For every family of sub-modules there is a finite subset of such that:
Examples
- Every finite module is Artinian.
- Every finitely generated module over an Artinian ring is Artinian.
- is not an Artinian module.
- 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.
- The examiner group as a module is Artinian, but not .
properties
- Every injective endomorphism is an automorphism
- For an exact sequence of modules are equivalent:
- is artinsch
- are artinsch
- For a (left) module over a (left) Artinian ring are equivalent:
- 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.
A ring is called rechtsartinsch if artinsch is a right module.
A ring is called Artinian when left and right are Arctic.
(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 .
- is right-noetherian, but neither leftartinian nor left-noetherian.
- 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 ).