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Difference between revisions of "Noetherian module"

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(Category:Associative rings and algebras)
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A [[Module|module]] for which every submodule has a finite system of generators. Equivalent conditions are: Every strictly ascending chain of submodules breaks off after finitely many terms; every non-empty set of submodules ordered by inclusion contains a maximal element. Submodules and quotient modules of a Noetherian module are Noetherian. If, in an [[Exact sequence|exact sequence]]
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A [[module]] for which every submodule has a finite system of generators. Equivalent conditions are: the ascending [[chain condition]] for [[submodule]]s (every strictly ascending chain of submodules breaks off after finitely many terms); every non-empty set of submodules ordered by inclusion contains a maximal element. Submodules and quotient modules of a Noetherian module are Noetherian. If, in an [[exact sequence]]
  
 
$$0\to M'\to M\to M''\to0,$$
 
$$0\to M'\to M\to M''\to0,$$
  
$M'$ and $M''$ are Noetherian, then so is $M$. A module over a [[Noetherian ring|Noetherian ring]] is Noetherian if and only if it is finitely generated.
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$M'$ and $M''$ are Noetherian, then so is $M$. A module over a [[Noetherian ring]] is Noetherian if and only if it is finitely generated.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR></table>
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[[Category:Associative rings and algebras]]

Revision as of 19:03, 25 October 2014

A module for which every submodule has a finite system of generators. Equivalent conditions are: the ascending chain condition for submodules (every strictly ascending chain of submodules breaks off after finitely many terms); every non-empty set of submodules ordered by inclusion contains a maximal element. Submodules and quotient modules of a Noetherian module are Noetherian. If, in an exact sequence

$$0\to M'\to M\to M''\to0,$$

$M'$ and $M''$ are Noetherian, then so is $M$. A module over a Noetherian ring is Noetherian if and only if it is finitely generated.

References

[1] S. Lang, "Algebra" , Addison-Wesley (1974)
How to Cite This Entry:
Noetherian module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_module&oldid=34018
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article