Noetherian ring

In algebra, certain structures (rings and modules ) are called noetherian if they can not contain infinite nesting of ever larger substructures. The term is named after the mathematician Emmy Noether.

  • 2.1 Examples
  • 2.2 Features

Noetherian modules

Let R be a unitary ring ( ie a ring with unit element). A left R- module M is called Noetherian if it satisfies one of the following equivalent conditions:

  • Each sub-module is finitely generated.
  • (Ascending chain condition) Every infinite ascending chain
  • ( Maximal condition for submodules ) Every non -empty set of R- submodules of M has a maximal element with respect to inclusion.

Examples

  • Every finite module is noetherian.
  • Every finitely generated module over a Noetherian ring is noetherschem.
  • Every finite direct sum of noetherian modules is noetherian.
  • Is not noetherian as a module.

Properties

  • Every surjective endomorphism is an automorphism.
  • For a short exact sequence are equivalent:
  • If V is a vector space, then V is Noetherian if and only if it is finite- dimensional. In this case, the module is also artinsch.
  • If R linksnoethersch, the Jacobson radical is nilpotent and semisimple, then R is also linksartinsch.
  • About a noetherschem ring every finitely generated module is finitely presented ( the converse is always).
  • The finitely generated modules over a noetherian ring form an abelian category; the condition that the ring is noetherian this is essential.

Noetherian rings

A ring R is called

  • Linksnoethersch if it is noetherian as a left R- module;
  • Rechtsnoethersch if it is Noetherian as R-module;
  • Noetherian if it is left and rechtsnoethersch.

For commutative rings all three terms are identical and equivalent to the fact that all ideals are finitely generated in R.

Examples

  • Is noetherian.
  • Quotients and localizations noetherian rings are noetherian.
  • Principal ideal rings, or generalized Dedekind rings are noetherian.
  • Is a Noetherian ring, so the polynomial ring is noetherian ( Hilbert basis theorem).
  • It follows that in general finitely generated algebras over a Noetherian ring are Noetherian again. In particular, finitely generated algebras over fields are noetherian.
  • The polynomial ring in infinitely many indeterminates is not Noetherian, since the ideal generated by all the indeterminates is not finitely generated.
  • The matrix ring is rechtsnoethersch, but neither linksartinsch still linksnoethersch.

Properties

  • Is in a ring, the zero ideal product maximal ideals, so the ring is Noetherian if and only if it is artinian.
  • In a noetherschem ring there are only finitely many minimal prime ideals.
Artinian ring Quotient ring Hilbert's basis theorem Noetherian topological space
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