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: ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle M}$

• Each sub-module is finitely generated.
• (Ascending chain condition) Any infinite ascending chain

${\displaystyle N_{1}\subseteq N_{2}\subseteq N_{3}\subseteq \dotsb }$

of sub-modules becomes stationary, i.e. i.e., there is an index so that ${\displaystyle n}$

${\displaystyle N_{n}=N_{n+1}=N_{n+2}=\dotsb }$

• (Maximum condition for sub-modules) Every non-empty set of sub-modules of has a maximum element regarding inclusion. ${\displaystyle M}$

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.
• ${\displaystyle \mathbb {Z} \left[{\tfrac {1}{p}}\right]}$is not more like a module. ${\displaystyle \mathbb {Z} }$

properties

• Every surjective endomorphism is an automorphism .
• For a short exact sequence are equivalent: ${\displaystyle \quad 0\rightarrow M_{1}\rightarrow M_{2}\rightarrow M_{3}\rightarrow 0}$

1. ${\displaystyle M_{2}}$ is noetherian.
2. ${\displaystyle M_{1},M_{3}}$ 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 . ${\displaystyle V}$ ${\displaystyle V}$
• If left- Noetherian, the Jacobson radical is nilpotent and semi - simple, then it is also left-Artinian . ${\displaystyle R}$ ${\displaystyle J=\operatorname {Rad} (R)}$ ${\displaystyle R/J}$ ${\displaystyle R}$
• 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 ${\displaystyle R}$

• linksnoetheric if it is noetherian as a left module; ${\displaystyle R}$
• rechtsnoethersch when used as is noetherian -Rechtsmodul; ${\displaystyle R}$
• 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. ${\displaystyle R}$

Examples

• Artin's rings are noetherian.
• ${\displaystyle \mathbb {Z} }$ 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 ). ${\displaystyle R}$ ${\displaystyle R[X]}$
• 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. ${\displaystyle \mathbb {C} [X_{1},X_{2},\ldots ]}$
• The matrix ring is right-noetheric, but neither left-arterial nor left-noetheric. ${\displaystyle {\begin{pmatrix}\mathbb {Z} &\mathbb {Q} \\0&\mathbb {Q} \end{pmatrix}}}$

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

• Noetherian room

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).