Artinian ring
The term artinian ring or artinian module (after Emil Artin ) describes a certain finiteness condition in the mathematical field of algebra. The term has some analogies to the notion of Noetherian ring, the two terms are not connected to each other in a very simple manner. For example, each artinian ring is noetherian, but not vice versa.
- 2.1 Definition
- 2.2 Examples
- 2.3 Properties
Artinian module
Definition
A module over a ring is called artinian if it satisfies one of the following equivalent conditions:
- Every non -empty set of submodules of has a minimal element with respect to inclusion.
- Each descending sequence of submodules becomes stationary, ie in a chain
- For each family of submodules of a finite subset exists, then applies that:
Examples
- Every finite module is artinian
- Every finitely generated module over an artinian ring is artinian
- Is not artinian module
- A finite direct sum of artinian modules is artinian
- If an (associative ) algebra over a field, and has a module finite - dimensional, so is artinian. For example, the rings and artinsch.
- Is artinian, but not
Properties
- Every injective endomorphism is an automorphism
- For an exact sequence of modules are equivalent:
- Are equivalent for a ( left ) module over a (left ) artinian ring: M is (left ) artinian
- M is (left ) Noetherian
- M is finitely generated
Artinian ring
Definition
A ring is called linksartinsch if it is artinian as a left - module.
A ring is called rechtsartinsch if it is artinian as right - module.
A ring is called artinian, is when the left and rechtsartinsch.
(Note: The submodules are then precisely the ( left / right ) ideals. )
Examples
- Bodies are artinian
- Let K be a field, R is a finitely generated K- algebra (ie for a suitable ideal), then R is an artinian ring if and only if.
- Is rechtsnoethersch, but neither linksartinsch still linksnoethersch.
- Is rechtsartinsch but not linksartinsch.
Properties
- An artinian ring is noetherian
- More specifically, a commutative ring with unit element if and only artinian if it is Noetherian and zero-dimensional (ie, if every prime ideal is a maximal ideal )
- An artinian ring integrity is already a body. It even applies the following stronger statement: An integral domain which satisfies the descending chain condition for principal ideals, is a body.
- Is in a ring, the zero ideal product maximal ideals, so the ring is artinian if and only if it is noetherian
- In an artinian ring, every prime ideal is already a maximum
- In an artinian ring, there are only finitely many maximal ideals (and thus only finitely many prime ideals )
- In an artinian ring, the nilradical is nilpotent
- Every artinian ring is a finite product of local artinian rings