Noetherian module

From Commalg
Jump to: navigation, search

This article defines a property of a module over a commutative unital ring

A commutative unital ring which satisfies this property as a module over itself, is termed a/an: Noetherian ring

Definition

Symbol-free definition

A module over a commutative unital ring is termed Noetherian if every submodule of it is finitely generated.

Relation with other properties

Weaker properties

Incomparable properties

Facts

  • A ring is Noetherian as a module over itself if and only if it is a Noetherian ring. This is because ideals of a ring are precisely its submodules.
  • Every finitely generated module over a Noetherian ring is a Noetherian module.

Metaproperties

Template:Submodule-closed

Any submodule of a Noetherian module is Noetherian.

Template:Finite-extension-closed

If a module has a Noetherian submodule and the quotient module is Noetherian, the module itself is Noetherian.

Template:Quotient-module-closed

Any quotient module of a Noetherian module is Noetherian.