# 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 .