Noetherian Ring
In algebra , certain structures ( rings and modules ) are called Noetherian if they can not contain an infinite nesting of ever larger substructures. The term is named after the mathematician Emmy Noether .
Noether's modules
Let it be a unitary ring (ie, a ring with one element). A - Links module is noetherian if it meets one of the following equivalent conditions:
- Each sub-module is finitely generated.
- (Ascending chain condition) Any infinite ascending chain
- of sub-modules becomes stationary, i.e. i.e., there is an index so that
- (Maximum condition for sub-modules) Every non-empty set of sub-modules of has a maximum element regarding inclusion.
Examples
- Every finite module is Noetherian.
- Every finitely generated module over a Noetherian ring is Noetherian.
- Every finite direct sum of Noetherian modules is Noetherian.
- is not more like a module.
properties
- Every surjective endomorphism is an automorphism .
- For a short exact sequence are equivalent:
- is noetherian.
- are noetherian.
- If a vector space is, then is Noetherian if and only if it is finite-dimensional. In this case the module is also Artinian .
- If left- Noetherian, the Jacobson radical is nilpotent and semi - simple, then it is also left-Artinian .
- Every finitely generated module is also finitely presented above a Noetherian ring (the reverse always applies).
- The finitely generated modules over a Noetherian ring form an Abelian category ; the requirement that the ring is noetherian is essential.
- Every real sub-module of a Noetherian module has a primary decomposition
Noether's rings
A ring is called
- linksnoetheric if it is noetherian as a left module;
- rechtsnoethersch when used as is noetherian -Rechtsmodul;
- noetherian when left and right noetherian .
With commutative rings, all three concepts are identical and equivalent to the fact that all ideals are finitely generated.
Examples
- Artin's rings are noetherian.
- is noetherian but not artinian.
- Quotients and localizations of Noetherian rings are Noetherian.
- Main ideal rings or more generally Dedekind rings are Noetherian.
- If a ring is Noetherian, the polynomial ring is also Noetherian ( Hilbert's basic theorem ).
- From this it follows that generally finitely generated algebras are again Noetherian over a Noetherian ring. In particular, finitely generated algebras over fields are Noetherian.
- The polynomial ring in infinitely many indeterminates is not Noetherian, since the ideal that is produced by all indeterminates is not finite.
- The matrix ring is right-noetheric, but neither left-arterial nor left-noetheric.
properties
- Every irreducible ideal in a Noetherian ring is a primary ideal .
- In a Noetherian ring, every real ideal can be represented as the intersection of a finite number of irreducible ideals. In particular, there is a primary decomposition in Noetherian rings
- In a Noetherian ring there are only a finite number of minimal prime ideals.
- Any non-unit in a Noetherian ring other than zero can be written as a finite product of irreducible elements. In particular, a Noetherian ring in which all irreducible elements are prime elements is a factorial ring .
- If the null ideal in a ring is the product of maximum ideals , then the ring is noetherian if and only if it is Artinian .
See also
literature
- Emmy Noether : Ideal theory in ring areas. In: Mathematical Annals . 83, 1921, pp. 24-66, GDZ
- Nicolas Bourbaki : Algèbre commutative. Volume 8/9: Chapitre 8: Dimension. Chapitre 9: Anneaux locaux noethériens complets. Masson, Paris 1983, ISBN 2-225-78716-6 ( Éléments de mathématique ).
- David Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry. Corrected 3rd printing. Springer-Verlag, New York NY 1999, ISBN 0-387-94268-8 ( Graduate Texts in Mathematics 150), (English).